A System of Four Generalized Sylvester Matrix Equations over the Quaternion Algebra

Abstract

In this paper, we make use of the simultaneous decomposition of eight quaternion matrices to study the solvability conditions and general solutions to a system of two-sided coupled Sylvester-type quaternion matrix equations $A_i{X}_i{C}_i+B_i{X}_{i+1}{D}_i={\Omega }_i,i=1,2,3,4.$ We design an algorithm to compute the general solution to the system and give a numerical example. Additionally, we consider the application of the system in the encryption and decryption of color images.

1. Introduction

1.1. Background

Quaternions, the decomposition of quaternion matrices and the matrix equations over quaternions, and so on, play important roles in mathematics and have a wide range of applications in various fields, such as color image processing, robotics, physics, aerospace engineering, control systems, and statistic model and graph theory. There are a great number of papers and monographs that investigate quaternion theory and corresponding applications from different aspects [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18].

Chen et al. [2] proposed a robust blind watermarking scheme based on quaternion QR decomposition for color image copyright protection. He et al. [19] considered the theory and application of a system of Sylvester-type quaternion matrix equations, the system shown as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{X}_1{A}_1-B_1{X}_2=C_1,\hfill \\ X_3{A}_2-B_2{X}_2=C_2,\hfill \\ X_3{A}_3-B_3{X}_4=C_3,\hfill \\ X_4{A}_4-B_4{X}_5=C_4,\hfill \\ X_6{A}_5-B_5{X}_5=C_5.\hfill \end{array}\right.\end{array}$$

Li et al. [11] came up with a quaternion biconjugate gradient method based on a structure-preserving method for solving non-Hermitian quaternion linear systems arising from color image deblurred problems. Took et al. [17] introduced the quaternion least mean square algorithm for the adaptive filtering of the three- and four-dimensional process.

There are a good deal of papers from various perspectives using various methods to study quaternion matrix equations, including the solvability conditions, general solutions, the properties of the general solution, the extreme rank of solutions, minimum norm least squares solution, $\varphi$-Hermitian solution, $\varphi$-skew-Hermitian solution, $\eta$-Hermitian solution, $\eta$-skew-Hermitian solution, and their applications [5,20,21,22,23,24,25,26,27,28,29,30,31,32].

Kyrchei [23] derived explicit formulas for the determinantal representations of solutions to the systems of the quaternion equations $A_{1}X=C_1$, $X{B}_2=C_2$ and $A_{1}X=C_1$, $A_{2}X=C_2$ using the determinantal representations of the Moore–Penrose matrix inverse put forward in [22]. Xu et al. [29] provided some useful necessary and sufficient conditions and a general solution to a constrained system of Sylvester-like matrix equations over the quaternion in terms of ranks and the Moore—Penrose inverse of the coefficient matrices. Xie et al. [33] considered the solvability conditions using ranks and the Moore–Penrose inverse for the system of three Sylvester-type quaternion matrix equations with ten variables $A_i{X}_i+Y_i{B}_i+C_i{Z}_i{D}_i+F_i{Z}_{i+1}{G}_i=E_i(i=1,2,3)$.

To our knowledge, there has been little information on the theory and applications of the following system:

$$\begin{array}{c}\hfill A_i{X}_i{C}_i+B_i{X}_{i+1}{D}_i={\Omega }_i,i=1,2,3,4.\end{array}$$

(1)

Motivated by the wide applications of quaternion matrix equations and the needs of their theoretical developments, we, in this paper, consider the solvability conditions, the general solutions and the applications to system (1).

The paper is organized as follows. First, we extend the simultaneous decomposition of seven quaternion matrices, which are shown in [21], to the simultaneous decomposition of eight quaternion matrices. Then, we make use of the simultaneous decomposition to prove that system (1) is consistent if and only if 40 rank equalities or 40 block matrix equalities hold. In the meantime, we also prove that these rank equalities as a whole are equivalent to these block matrix equalities as a whole. Next, we show an algorithm which clearly illustrates the steps taken to obtain the general solution to system (1) and we also give a numerical example. Afterwards, we make use of the system of two-sided coupled Sylvester-type quaternion matrix equations to develop a framework that can be used to encrypt and decrypt four color images simultaneously. Finally, we summarize our work.

1.2. Notation

Let $\mathbb{R}$ and $\mathbb{H}$ stand for the real number field and quaternion algebra, respectively. $\mathbb{H}$ can be viewed as a four-dimensional linear space over $\mathbb{R}$ with the basis: $\{1,\mathbf{i},\mathbf{j},\mathbf{k}\}$, satisfying ${\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=\mathbf{i}\mathbf{j}\mathbf{k}=-1$, $\mathbf{i}\mathbf{j}=-\mathbf{j}\mathbf{i}=\mathbf{k}$, $\mathbf{j}\mathbf{k}=-\mathbf{k}\mathbf{j}=\mathbf{i}$ and $\mathbf{k}\mathbf{i}=-\mathbf{i}\mathbf{k}=\mathbf{j}$, where $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are called imaginary units.

Throughout this paper, we denote ${\mathbb{H}}^{m\times n}$ as all $m\times n$ matrices over the real quaternions. The rank of a quaternion matrix A over $\mathbb{H}$ is defined to be the maximum number of columns of A, which are linearly independent to the right. Quaternion matrix A and $PAQ$ have the same rank if P and Q are invertible quaternion matrices [18]. For convenience, we use $r_{A_{11}{A}_{12}\cdots A_{1n}|A_{21}{A}_{22}\cdots A_{2n}|A_{m1}{A}_{m2}\cdots A_{mn}}$ to represent the rank of a block quaternion matrix, as follows:

$$\begin{array}{c}\hfill \left(\begin{array}{cccc}{A}_{11}& A_{12}& \cdots & A_{1n}\\ A_{21}& A_{22}& \cdots & A_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ A_{m1}& A_{m2}& \cdots & A_{mn}\end{array}\right),\end{array}$$

(2)

and if $A_{ij}$ in the block (2) is a zero matrix, we use “0” to represent it. For instance, $r_{A_i{\Omega }_i{B}_i}$, $r_{C_i\left|{\Omega }_i\right|D_i}$ and $r_{A_i{\Omega }_i|0D_i}$ stand for the ranks of the following block matrices:

$$\begin{array}{c}\hfill (A_i,{\Omega }_i,B_i),\left(\begin{array}{c}{C}_i\\ {\Omega }_i\\ D_i\end{array}\right),\left(\begin{array}{cc}{A}_i& {\Omega }_i\\ & D_i\end{array}\right),\end{array}$$

respectively.

2. The Solvability Conditions of System (1)

In this section, we investigate the solvability conditions of the system of quaternion matrix equations as follows:

$$\begin{array}{c}\hfill A_i{X}_i{C}_i+B_i{X}_{i+1}{D}_i={\Omega }_i,i=1,2,3,4.\end{array}$$

Notice that the sizes of the coefficient matrices $A_i$, $B_i$, $C_i$ and $D_i$ have certain rules. They can be arranged into block matrices as

$$\begin{array}{cc}& \begin{array}{ccccc}{q}_1& q_2& q_3& q_4& q_5\end{array}\\ \begin{array}{c}{p}_1\\ p_2\\ p_3\\ p_4\end{array}& \left(\begin{array}{ccccc}{A}_1& B_1& & & \\ & A_2& B_2& & \\ & & A_3& B_3& \\ & & & A_4& B_4\end{array}\right)\end{array}$$

(3)

and

$$\begin{array}{cc}& \begin{array}{cccc}{s}_1& s_2& s_3& s_4\end{array}\\ \begin{array}{c}{t}_1\\ t_2\\ t_3\\ t_4\\ t_5\end{array}& \left(\begin{array}{cccc}{C}_1& & & \\ D_1& C_2& & \\ & D_2& C_3& \\ & & D_3& C_4\\ & & & D_4\end{array}\right)\end{array}$$

(4)

 Lemma 1 

([19,21]). Considering block matrix (3), there are nonsingular quaternion matrices $P_i\in {\mathbb{H}}^{p_i\times p_i},Q_i\in {\mathbb{H}}^{q_i\times q_i}$, such that

$$\begin{array}{c}\hfill P_i{A}_i{Q}_i=S_{a_i},P_i{B}_i{Q}_{i+1}=S_{b_i},i=1,\cdots ,4,\end{array}$$

(5)

where $\begin{array}{cc}& \begin{array}{ccc}{q}_1& q_2& q_3\end{array}\\ \begin{array}{c}{p}_1\\ p_2\end{array}& \left(\begin{array}{ccc}{S}_{a1}& S_{b1}& \\ & S_{a2}& S_{b2}\end{array}\right)\end{array}$, denoted by $\widehat{A{B}_1}$, $\begin{array}{cc}& \begin{array}{ccc}{q}_3& q_4& q_5\end{array}\\ \begin{array}{c}{p}_3\\ p_4\end{array}& \left(\begin{array}{ccc}{S}_{a3}& S_{b3}& \\ & S_{a4}& S_{b4}\end{array}\right)\end{array}$, denoted by $\widehat{A{B}_2}$, and we have

$$\hfill \widehat{A{B}_1}=\left(\begin{array}{ccc}\begin{array}{c}\begin{array}{ccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& I& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& I& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}& \begin{array}{c}\begin{array}{ccccccccccccccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 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0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}& \begin{array}{c}\begin{array}{ccccccccccccccccccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 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$$\begin{array}{c}\hfill \widehat{A{B}_2}=\left(\begin{array}{ccc}\begin{array}{c}\begin{array}{ccccccccccccccccccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}& \begin{array}{c}\begin{array}{ccccccccccccccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}& \\ & \begin{array}{c}\begin{array}{ccccccccccccccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& I& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}& \begin{array}{c}\begin{array}{ccccccccc}I& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& I& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& I& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& I& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}\end{array}\right),\end{array}$$

where I and 0 stand for the identity matrix and zero matrix with appropriate size, respectively. Similarly, there are the nonsingular quaternion matrices $T_i\in {\mathbb{H}}^{t_i\times t_i},S_i\in {\mathbb{H}}^{s_i\times s_i}$, such that

$$\begin{array}{c}\hfill T_i{C}_i{S}_i=S_{c_i},T_{i+1}{D}_i{S}_i=S_{d_i},i=1,\cdots ,4,\end{array}$$

(6)

where

$$\begin{array}{c}\hfill {\left(\begin{array}{cccc}{S}_{c_1}& & & \\ S_{d_1}& S_{c_2}& & \\ & S_{d_2}& S_{c_3}& \\ & & S_{d_3}& S_{c_4}\\ & & & S_{d_4}\end{array}\right)}^T=\left(\begin{array}{ccccc}{S}_{a_1}& S_{b_1}& & & \\ & S_{a_2}& S_{b_2}& & \\ & & S_{a_3}& S_{b_3}& \\ & & & S_{a_4}& S_{b_4}\end{array}\right).\end{array}$$

He et al. [21] showed the simultaneous decomposition of seven quaternion matrices. Starting from that, and using a similar method as [19], we can reach our conclusion.

According to (5) and (6), we have

$$\begin{array}{c}\hfill P_i^{-1}{S}_{a_i}{Q}_i^{-1}{X}_i{T}_i^{-1}{S}_{c_i}{S}_i^{-1}+P_i^{-1}{S}_{b_i}{Q}_{i+1}^{-1}{X}_{i+1}{T}_{i+1}^{-1}{S}_{d_i}{S}_i^{-1}={\Omega }_i,i=1,2,3,4.\end{array}$$

Multiply $P_i$ and $S_i$ to the left and right sides of the equations, respectively. Then, we have

$$\begin{array}{c}\hfill S_{a_i}{Q}_i^{-1}{X}_i{T}_i^{-1}{S}_{c_i}+S_{b_i}{Q}_{i+1}^{-1}{X}_{i+1}{T}_{i+1}^{-1}{S}_{d_i}=P_i{\Omega }_i{S}_i,i=1,2,3,4.\end{array}$$

Let $\widehat{{\Omega }_i}=P_i{\Omega }_i{S}_i,i=1,\cdots ,4$, $\widehat{X_j}=Q_j^{-1}{X}_j{T}_j^{-1},j=1,\cdots ,5$. Hence, we have

$$\begin{array}{c}\hfill S_{a_i}\widehat{X_i}{S}_{c_i}+S_{b_i}\widehat{X_{i+1}}{S}_{d_i}=\widehat{{\Omega }_i},i=1,2,3,4.\end{array}$$

(7)

According to the equivalence canonical forms of (3) and (4) and system (7), we divide $\widehat{X_j},j=1,\cdots ,5$ and $\widehat{{\Omega }_i},i=1,2,3,4$ into the following partitioned matrices:

$$\begin{array}{cc}\hfill & \widehat{X_1}={\left(X_{ij}^{\left(1\right)}\right)}_{9\times 9},\widehat{X_2}={\left(X_{ij}^{\left(2\right)}\right)}_{21\times 21},\widehat{X_3}={\left(X_{ij}^{\left(3\right)}\right)}_{25\times 25},\widehat{X_4}={\left(X_{ij}^{\left(4\right)}\right)}_{21\times 21},\widehat{X_5}={\left(X_{ij}^{\left(5\right)}\right)}_{9\times 9},\hfill \\ \hfill & \widehat{{\Omega }_1}={\left({\omega }_{ij}^{\left(1\right)}\right)}_{16\times 16},\widehat{{\Omega }_2}={\left({\omega }_{ij}^{\left(2\right)}\right)}_{24\times 24},\widehat{{\Omega }_3}={\left({\omega }_{ij}^{\left(3\right)}\right)}_{24\times 24},\widehat{{\Omega }_4}={\left({\omega }_{ij}^{\left(4\right)}\right)}_{16\times 16}.\hfill \end{array}$$

Through computation, we have

$$\begin{array}{c}\hfill \widehat{{\Omega }_1}=({\Phi }_1^{\left(1\right)},{\Phi }_2^{\left(1\right)}),\end{array}$$

(8)

where

$$\begin{array}{c}\hfill {\Phi }_1^{\left(1\right)}=\left(\begin{array}{cccccc}{X}_{11}^{\left(1\right)}+X_{11}^{\left(2\right)}& X_{12}^{\left(1\right)}+X_{12}^{\left(2\right)}& X_{13}^{\left(1\right)}+X_{13}^{\left(2\right)}& X_{14}^{\left(1\right)}+X_{14}^{\left(2\right)}& X_{15}^{\left(1\right)}+X_{15}^{\left(2\right)}& X_{16}^{\left(1\right)}+X_{16}^{\left(2\right)}\\ X_{21}^{\left(1\right)}+X_{21}^{\left(2\right)}& X_{22}^{\left(1\right)}+X_{22}^{\left(2\right)}& X_{23}^{\left(1\right)}+X_{23}^{\left(2\right)}& X_{24}^{\left(1\right)}+X_{24}^{\left(2\right)}& X_{25}^{\left(1\right)}+X_{25}^{\left(2\right)}& X_{26}^{\left(1\right)}+X_{26}^{\left(2\right)}\\ X_{31}^{\left(1\right)}+X_{31}^{\left(2\right)}& X_{32}^{\left(1\right)}+X_{32}^{\left(2\right)}& X_{33}^{\left(1\right)}+X_{33}^{\left(2\right)}& X_{34}^{\left(1\right)}+X_{34}^{\left(2\right)}& X_{35}^{\left(1\right)}+X_{35}^{\left(2\right)}& X_{36}^{\left(1\right)}+X_{36}^{\left(2\right)}\\ X_{41}^{\left(1\right)}+X_{41}^{\left(2\right)}& X_{42}^{\left(1\right)}+X_{42}^{\left(2\right)}& X_{43}^{\left(1\right)}+X_{43}^{\left(2\right)}& X_{44}^{\left(1\right)}+X_{44}^{\left(2\right)}& X_{45}^{\left(1\right)}+X_{45}^{\left(2\right)}& X_{46}^{\left(1\right)}+X_{46}^{\left(2\right)}\\ X_{51}^{\left(1\right)}+X_{51}^{\left(2\right)}& X_{52}^{\left(1\right)}+X_{52}^{\left(2\right)}& X_{53}^{\left(1\right)}+X_{53}^{\left(2\right)}& X_{54}^{\left(1\right)}+X_{54}^{\left(2\right)}& X_{55}^{\left(1\right)}+X_{55}^{\left(2\right)}& X_{56}^{\left(1\right)}+X_{56}^{\left(2\right)}\\ X_{61}^{\left(1\right)}+X_{61}^{\left(2\right)}& X_{62}^{\left(1\right)}+X_{62}^{\left(2\right)}& X_{63}^{\left(1\right)}+X_{63}^{\left(2\right)}& X_{64}^{\left(1\right)}+X_{64}^{\left(2\right)}& X_{65}^{\left(1\right)}+X_{65}^{\left(2\right)}& X_{66}^{\left(1\right)}+X_{66}^{\left(2\right)}\\ X_{71}^{\left(1\right)}+X_{71}^{\left(2\right)}& X_{72}^{\left(1\right)}+X_{72}^{\left(2\right)}& X_{73}^{\left(1\right)}+X_{73}^{\left(2\right)}& X_{74}^{\left(1\right)}+X_{74}^{\left(2\right)}& X_{75}^{\left(1\right)}+X_{75}^{\left(2\right)}& X_{76}^{\left(1\right)}+X_{76}^{\left(2\right)}\\ X_{81}^{\left(1\right)}& X_{82}^{\left(1\right)}& X_{83}^{\left(1\right)}& X_{84}^{\left(1\right)}& X_{85}^{\left(1\right)}& X_{86}^{\left(1\right)}\\ X_{81}^{\left(2\right)}& X_{82}^{\left(2\right)}& X_{83}^{\left(2\right)}& X_{84}^{\left(2\right)}& X_{85}^{\left(2\right)}& X_{86}^{\left(2\right)}\\ X_{91}^{\left(2\right)}& X_{92}^{\left(2\right)}& X_{93}^{\left(2\right)}& X_{94}^{\left(2\right)}& X_{95}^{\left(2\right)}& X_{96}^{\left(2\right)}\\ X_{10,1}^{\left(2\right)}& X_{10,2}^{\left(2\right)}& X_{10,3}^{\left(2\right)}& X_{10,4}^{\left(2\right)}& X_{10,5}^{\left(2\right)}& X_{10,6}^{\left(2\right)}\\ X_{11,1}^{\left(2\right)}& X_{11,2}^{\left(2\right)}& X_{11,3}^{\left(2\right)}& X_{11,4}^{\left(2\right)}& X_{11,5}^{\left(2\right)}& X_{11,6}^{\left(2\right)}\\ X_{12,1}^{\left(2\right)}& X_{12,2}^{\left(2\right)}& X_{12,3}^{\left(2\right)}& X_{12,4}^{\left(2\right)}& X_{12,5}^{\left(2\right)}& X_{12,6}^{\left(2\right)}\\ X_{13,1}^{\left(2\right)}& X_{13,2}^{\left(2\right)}& X_{13,3}^{\left(2\right)}& X_{13,4}^{\left(2\right)}& X_{13,5}^{\left(2\right)}& X_{13,6}^{\left(2\right)}\\ X_{14,1}^{\left(2\right)}& X_{14,2}^{\left(2\right)}& X_{14,3}^{\left(2\right)}& X_{14,4}^{\left(2\right)}& X_{14,5}^{\left(2\right)}& X_{14,6}^{\left(2\right)}\\ 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\Phi }_2^{\left(1\right)}=\left(\begin{array}{cccccccccc}{X}_{17}^{\left(1\right)}+X_{17}^{\left(2\right)}& X_{18}^{\left(1\right)}& X_{18}^{\left(2\right)}& X_{19}^{\left(2\right)}& X_{1,10}^{\left(2\right)}& X_{1,11}^{\left(2\right)}& X_{1,12}^{\left(2\right)}& X_{1,13}^{\left(2\right)}& X_{1,14}^{\left(2\right)}& 0\\ X_{27}^{\left(1\right)}+X_{27}^{\left(2\right)}& X_{28}^{\left(1\right)}& X_{28}^{\left(2\right)}& X_{29}^{\left(2\right)}& X_{2,10}^{\left(2\right)}& X_{2,11}^{\left(2\right)}& X_{2,12}^{\left(2\right)}& X_{2,13}^{\left(2\right)}& X_{2,14}^{\left(2\right)}& 0\\ X_{37}^{\left(1\right)}+X_{37}^{\left(2\right)}& X_{38}^{\left(1\right)}& X_{38}^{\left(2\right)}& X_{39}^{\left(2\right)}& X_{3,10}^{\left(2\right)}& X_{3,11}^{\left(2\right)}& X_{3,12}^{\left(2\right)}& X_{3,13}^{\left(2\right)}& X_{3,14}^{\left(2\right)}& 0\\ X_{47}^{\left(1\right)}+X_{47}^{\left(2\right)}& X_{48}^{\left(1\right)}& X_{48}^{\left(2\right)}& X_{49}^{\left(2\right)}& X_{4,10}^{\left(2\right)}& X_{4,11}^{\left(2\right)}& X_{4,12}^{\left(2\right)}& X_{4,13}^{\left(2\right)}& X_{4,14}^{\left(2\right)}& 0\\ X_{57}^{\left(1\right)}+X_{57}^{\left(2\right)}& X_{58}^{\left(1\right)}& X_{58}^{\left(2\right)}& X_{59}^{\left(2\right)}& X_{5,10}^{\left(2\right)}& X_{5,11}^{\left(2\right)}& X_{5,12}^{\left(2\right)}& X_{5,13}^{\left(2\right)}& X_{5,14}^{\left(2\right)}& 0\\ X_{67}^{\left(1\right)}+X_{67}^{\left(2\right)}& X_{68}^{\left(1\right)}& X_{68}^{\left(2\right)}& X_{69}^{\left(2\right)}& X_{6,10}^{\left(2\right)}& X_{6,11}^{\left(2\right)}& X_{6,12}^{\left(2\right)}& X_{6,13}^{\left(2\right)}& X_{6,14}^{\left(2\right)}& 0\\ X_{77}^{\left(1\right)}+X_{77}^{\left(2\right)}& X_{78}^{\left(1\right)}& X_{78}^{\left(2\right)}& X_{79}^{\left(2\right)}& X_{7,10}^{\left(2\right)}& X_{7,11}^{\left(2\right)}& X_{7,12}^{\left(2\right)}& X_{7,13}^{\left(2\right)}& X_{7,14}^{\left(2\right)}& 0\\ X_{87}^{\left(1\right)}& X_{88}^{\left(1\right)}& 0& 0& 0& 0& 0& 0& 0& 0\\ X_{87}^{\left(2\right)}& 0& X_{88}^{\left(2\right)}& X_{89}^{\left(2\right)}& X_{8,10}^{\left(2\right)}& X_{8,11}^{\left(2\right)}& X_{8,12}^{\left(2\right)}& X_{8,13}^{\left(2\right)}& X_{8,14}^{\left(2\right)}& 0\\ X_{97}^{\left(2\right)}& 0& X_{98}^{\left(2\right)}& X_{99}^{\left(2\right)}& X_{9,10}^{\left(2\right)}& X_{9,11}^{\left(2\right)}& X_{9,12}^{\left(2\right)}& X_{9,13}^{\left(2\right)}& X_{9,14}^{\left(2\right)}& 0\\ X_{10,7}^{\left(2\right)}& 0& X_{10,8}^{\left(2\right)}& X_{10,9}^{\left(2\right)}& X_{10,10}^{\left(2\right)}& X_{10,11}^{\left(2\right)}& X_{10,12}^{\left(2\right)}& X_{10,13}^{\left(2\right)}& X_{10,14}^{\left(2\right)}& 0\\ X_{11,7}^{\left(2\right)}& 0& X_{11,8}^{\left(2\right)}& X_{11,9}^{\left(2\right)}& X_{11,10}^{\left(2\right)}& X_{11,11}^{\left(2\right)}& X_{11,12}^{\left(2\right)}& X_{11,13}^{\left(2\right)}& X_{11,14}^{\left(2\right)}& 0\\ X_{12,7}^{\left(2\right)}& 0& X_{12,8}^{\left(2\right)}& X_{12,9}^{\left(2\right)}& X_{12,10}^{\left(2\right)}& X_{12,11}^{\left(2\right)}& X_{12,12}^{\left(2\right)}& X_{12,13}^{\left(2\right)}& X_{12,14}^{\left(2\right)}& 0\\ X_{13,7}^{\left(2\right)}& 0& X_{13,8}^{\left(2\right)}& X_{13,9}^{\left(2\right)}& X_{13,10}^{\left(2\right)}& X_{13,11}^{\left(2\right)}& X_{13,12}^{\left(2\right)}& X_{13,13}^{\left(2\right)}& X_{13,14}^{\left(2\right)}& 0\\ X_{14,7}^{\left(2\right)}& 0& X_{14,8}^{\left(2\right)}& X_{14,9}^{\left(2\right)}& X_{14,10}^{\left(2\right)}& X_{14,11}^{\left(2\right)}& X_{14,12}^{\left(2\right)}& X_{14,13}^{\left(2\right)}& X_{14,14}^{\left(2\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right).\end{array}$$

We have

$$\begin{array}{c}\hfill \widehat{{\Omega }_2}=\left(\begin{array}{ccc}{\Phi }_{11}^{\left(2\right)}& {\Phi }_{12}^{\left(2\right)}& {\Phi }_{13}^{\left(2\right)}\\ {\Phi }_{21}^{\left(2\right)}& {\Phi }_{22}^{\left(2\right)}& {\Phi }_{23}^{\left(2\right)}\end{array}\right),\end{array}$$

(9)

where

$$\begin{array}{cc}\hfill {\Phi }_{11}^{\left(2\right)}=& \left(\begin{array}{cccccccc}{X}_{11}^{\left(2\right)}+X_{11}^{\left(3\right)}& X_{12}^{\left(2\right)}+X_{12}^{\left(3\right)}& X_{13}^{\left(2\right)}+X_{13}^{\left(3\right)}& X_{14}^{\left(2\right)}+X_{14}^{\left(3\right)}& X_{15}^{\left(2\right)}+X_{15}^{\left(3\right)}& X_{16}^{\left(2\right)}& X_{18}^{\left(2\right)}+X_{16}^{\left(3\right)}& X_{19}^{\left(2\right)}+X_{17}^{\left(3\right)}\\ X_{21}^{\left(2\right)}+X_{21}^{\left(3\right)}& X_{22}^{\left(2\right)}+X_{22}^{\left(3\right)}& X_{23}^{\left(2\right)}+X_{23}^{\left(3\right)}& X_{24}^{\left(2\right)}+X_{24}^{\left(3\right)}& X_{25}^{\left(2\right)}+X_{25}^{\left(3\right)}& X_{26}^{\left(2\right)}& X_{28}^{\left(2\right)}+X_{26}^{\left(3\right)}& X_{29}^{\left(2\right)}+X_{27}^{\left(3\right)}\\ X_{31}^{\left(2\right)}+X_{31}^{\left(3\right)}& X_{32}^{\left(2\right)}+X_{32}^{\left(3\right)}& X_{33}^{\left(2\right)}+X_{33}^{\left(3\right)}& X_{34}^{\left(2\right)}+X_{34}^{\left(3\right)}& X_{35}^{\left(2\right)}+X_{35}^{\left(3\right)}& X_{36}^{\left(2\right)}& X_{38}^{\left(2\right)}+X_{36}^{\left(3\right)}& X_{39}^{\left(2\right)}+X_{37}^{\left(3\right)}\\ X_{41}^{\left(2\right)}+X_{41}^{\left(3\right)}& X_{42}^{\left(2\right)}+X_{42}^{\left(3\right)}& X_{43}^{\left(2\right)}+X_{43}^{\left(3\right)}& X_{44}^{\left(2\right)}+X_{44}^{\left(3\right)}& X_{45}^{\left(2\right)}+X_{45}^{\left(3\right)}& X_{46}^{\left(2\right)}& X_{48}^{\left(2\right)}+X_{46}^{\left(3\right)}& X_{49}^{\left(2\right)}+X_{47}^{\left(3\right)}\\ X_{51}^{\left(2\right)}+X_{51}^{\left(3\right)}& X_{52}^{\left(2\right)}+X_{52}^{\left(3\right)}& X_{53}^{\left(2\right)}+X_{53}^{\left(3\right)}& X_{54}^{\left(2\right)}+X_{54}^{\left(3\right)}& X_{55}^{\left(2\right)}+X_{55}^{\left(3\right)}& X_{56}^{\left(2\right)}& X_{58}^{\left(2\right)}+X_{56}^{\left(3\right)}& X_{59}^{\left(2\right)}+X_{57}^{\left(3\right)}\\ X_{61}^{\left(2\right)}& X_{62}^{\left(2\right)}& X_{63}^{\left(2\right)}& X_{64}^{\left(2\right)}& X_{65}^{\left(2\right)}& X_{66}^{\left(2\right)}& X_{68}^{\left(2\right)}& X_{69}^{\left(2\right)}\\ X_{81}^{\left(2\right)}+X_{61}^{\left(3\right)}& X_{82}^{\left(2\right)}+X_{62}^{\left(3\right)}& X_{83}^{\left(2\right)}+X_{63}^{\left(3\right)}& X_{84}^{\left(2\right)}+X_{64}^{\left(3\right)}& X_{85}^{\left(2\right)}+X_{65}^{\left(3\right)}& X_{86}^{\left(2\right)}& X_{88}^{\left(2\right)}+X_{66}^{\left(3\right)}& X_{89}^{\left(2\right)}+X_{67}^{\left(3\right)}\\ X_{91}^{\left(2\right)}+X_{71}^{\left(3\right)}& X_{92}^{\left(2\right)}+X_{72}^{\left(3\right)}& X_{93}^{\left(2\right)}+X_{73}^{\left(3\right)}& X_{94}^{\left(2\right)}+X_{74}^{\left(3\right)}& X_{95}^{\left(2\right)}+X_{75}^{\left(3\right)}& X_{96}^{\left(2\right)}& X_{98}^{\left(2\right)}+X_{76}^{\left(3\right)}& X_{99}^{\left(2\right)}+X_{77}^{\left(3\right)}\\ X_{10,1}^{\left(2\right)}+X_{81}^{\left(3\right)}& X_{10,2}^{\left(2\right)}+X_{82}^{\left(3\right)}& X_{10,3}^{\left(2\right)}+X_{83}^{\left(3\right)}& X_{10,4}^{\left(2\right)}+X_{84}^{\left(3\right)}& X_{10,5}^{\left(2\right)}+X_{85}^{\left(3\right)}& X_{10,6}^{\left(2\right)}& X_{10,8}^{\left(2\right)}+X_{86}^{\left(3\right)}& X_{10,9}^{\left(2\right)}+X_{87}^{\left(3\right)}\\ X_{11,1}^{\left(2\right)}+X_{91}^{\left(3\right)}& X_{11,2}^{\left(2\right)}+X_{92}^{\left(3\right)}& X_{11,3}^{\left(2\right)}+X_{93}^{\left(3\right)}& X_{11,4}^{\left(2\right)}+X_{94}^{\left(3\right)}& X_{11,5}^{\left(2\right)}+X_{95}^{\left(3\right)}& X_{11,6}^{\left(2\right)}& X_{11,8}^{\left(2\right)}+X_{9,6}^{\left(3\right)}& X_{11,9}^{\left(2\right)}+X_{97}^{\left(3\right)}\\ X_{12,1}^{\left(2\right)}+X_{10,1}^{\left(3\right)}& X_{12,2}^{\left(2\right)}+X_{10,2}^{\left(3\right)}& X_{12,3}^{\left(2\right)}+X_{10,3}^{\left(3\right)}& X_{12,4}^{\left(2\right)}+X_{10,4}^{\left(3\right)}& X_{12,5}^{\left(2\right)}+X_{10,5}^{\left(3\right)}& X_{12,6}^{\left(2\right)}& X_{12,8}^{\left(2\right)}+X_{10,6}^{\left(3\right)}& X_{12,9}^{\left(2\right)}+X_{10,7}^{\left(3\right)}\\ X_{13,1}^{\left(2\right)}& X_{13,2}^{\left(2\right)}& X_{13,3}^{\left(2\right)}& X_{13,4}^{\left(2\right)}& X_{13,5}^{\left(2\right)}& X_{13,6}^{\left(2\right)}& X_{13,8}^{\left(2\right)}& X_{13,9}^{\left(2\right)}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\Phi }_{21}^{\left(2\right)}=& \left(\begin{array}{cccccccc}{X}_{15,1}^{\left(2\right)}+X_{11,1}^{\left(3\right)}& X_{15,2}^{\left(2\right)}+X_{11,2}^{\left(3\right)}& X_{15,3}^{\left(2\right)}+X_{11,3}^{\left(3\right)}& X_{15,4}^{\left(2\right)}+X_{11,4}^{\left(3\right)}& X_{15,5}^{\left(2\right)}+X_{11,5}^{\left(3\right)}& X_{15,6}^{\left(2\right)}& X_{15,8}^{\left(2\right)}+X_{11,6}^{\left(3\right)}& X_{15,9}^{\left(2\right)}+X_{11,7}^{\left(3\right)}\\ X_{16,1}^{\left(2\right)}+X_{12,1}^{\left(3\right)}& X_{16,2}^{\left(2\right)}+X_{12,2}^{\left(3\right)}& X_{16,3}^{\left(2\right)}+X_{12,3}^{\left(3\right)}& X_{16,4}^{\left(2\right)}+X_{12,4}^{\left(3\right)}& X_{16,5}^{\left(2\right)}+X_{12,5}^{\left(3\right)}& X_{16,6}^{\left(2\right)}& X_{16,8}^{\left(2\right)}+X_{12,6}^{\left(3\right)}& X_{16,9}^{\left(2\right)}+X_{12,7}^{\left(3\right)}\\ X_{17,1}^{\left(2\right)}+X_{13,1}^{\left(3\right)}& X_{17,2}^{\left(2\right)}+X_{13,2}^{\left(3\right)}& X_{17,3}^{\left(2\right)}+X_{13,3}^{\left(3\right)}& X_{17,4}^{\left(2\right)}+X_{13,4}^{\left(3\right)}& X_{17,5}^{\left(2\right)}+X_{13,5}^{\left(3\right)}& X_{17,6}^{\left(2\right)}& X_{17,8}^{\left(2\right)}+X_{13,6}^{\left(3\right)}& X_{17,9}^{\left(2\right)}+X_{13,7}^{\left(3\right)}\\ X_{18,1}^{\left(2\right)}+X_{14,1}^{\left(3\right)}& X_{18,2}^{\left(2\right)}+X_{14,2}^{\left(3\right)}& X_{18,3}^{\left(2\right)}+X_{14,3}^{\left(3\right)}& X_{18,4}^{\left(2\right)}+X_{14,4}^{\left(3\right)}& X_{18,5}^{\left(2\right)}+X_{14,5}^{\left(3\right)}& X_{18,6}^{\left(2\right)}& X_{18,8}^{\left(2\right)}+X_{14,6}^{\left(3\right)}& X_{18,9}^{\left(2\right)}+X_{14,7}^{\left(3\right)}\\ X_{19,1}^{\left(2\right)}+X_{15,1}^{\left(3\right)}& X_{19,2}^{\left(2\right)}+X_{15,2}^{\left(3\right)}& X_{19,3}^{\left(2\right)}+X_{15,3}^{\left(3\right)}& X_{19,4}^{\left(2\right)}+X_{15,4}^{\left(3\right)}& X_{19,5}^{\left(2\right)}+X_{15,5}^{\left(3\right)}& X_{19,6}^{\left(2\right)}& X_{19,8}^{\left(2\right)}+X_{15,6}^{\left(3\right)}& X_{19,9}^{\left(2\right)}+X_{15,7}^{\left(3\right)}\\ X_{20,1}^{\left(2\right)}& X_{20,2}^{\left(2\right)}& X_{20,3}^{\left(2\right)}& X_{20,4}^{\left(2\right)}& X_{20,5}^{\left(2\right)}& X_{20,6}^{\left(2\right)}& X_{20,8}^{\left(2\right)}& X_{20,9}^{\left(2\right)}\\ X_{16,1}^{\left(3\right)}& X_{16,2}^{\left(3\right)}& X_{16,3}^{\left(3\right)}& X_{16,4}^{\left(3\right)}& X_{16,5}^{\left(3\right)}& 0& X_{16,6}^{\left(3\right)}& X_{16,7}^{\left(3\right)}\\ X_{17,1}^{\left(3\right)}& X_{17,2}^{\left(3\right)}& X_{17,3}^{\left(3\right)}& X_{17,4}^{\left(3\right)}& X_{17,5}^{\left(3\right)}& 0& X_{17,6}^{\left(3\right)}& X_{17,7}^{\left(3\right)}\\ X_{18,1}^{\left(3\right)}& X_{18,2}^{\left(3\right)}& X_{18,3}^{\left(3\right)}& X_{18,4}^{\left(3\right)}& X_{18,5}^{\left(3\right)}& 0& X_{18,6}^{\left(3\right)}& X_{18,7}^{\left(3\right)}\\ X_{19,1}^{\left(3\right)}& X_{19,2}^{\left(3\right)}& X_{19,3}^{\left(3\right)}& X_{19,4}^{\left(3\right)}& X_{19,5}^{\left(3\right)}& 0& X_{19,6}^{\left(3\right)}& X_{19,7}^{\left(3\right)}\\ X_{20,1}^{\left(3\right)}& X_{20,2}^{\left(3\right)}& X_{20,3}^{\left(3\right)}& X_{20,4}^{\left(3\right)}& X_{20,5}^{\left(3\right)}& 0& X_{20,6}^{\left(3\right)}& X_{20,7}^{\left(3\right)}\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill {\Phi }_{12}^{\left(2\right)}=\left(\begin{array}{ccccccc}{X}_{1,10}^{\left(2\right)}+X_{18}^{\left(3\right)}& X_{1,11}^{\left(2\right)}+X_{19}^{\left(3\right)}& X_{1,12}^{\left(2\right)}+X_{1,10}^{\left(3\right)}& X_{1,13}^{\left(2\right)}& X_{1,15}^{\left(2\right)}+X_{1,11}^{\left(3\right)}& X_{1,16}^{\left(2\right)}+X_{1,12}^{\left(3\right)}& X_{1,17}^{\left(2\right)}+X_{1,13}^{\left(3\right)}\\ X_{2,10}^{\left(2\right)}+X_{28}^{\left(3\right)}& X_{2,11}^{\left(2\right)}+X_{29}^{\left(3\right)}& X_{2,12}^{\left(2\right)}+X_{2,10}^{\left(3\right)}& X_{2,13}^{\left(2\right)}& X_{2,15}^{\left(2\right)}+X_{2,11}^{\left(3\right)}& X_{2,16}^{\left(2\right)}+X_{2,12}^{\left(3\right)}& X_{2,17}^{\left(2\right)}+X_{2,13}^{\left(3\right)}\\ X_{3,10}^{\left(2\right)}+X_{38}^{\left(3\right)}& X_{3,11}^{\left(2\right)}+X_{39}^{\left(3\right)}& X_{3,12}^{\left(2\right)}+X_{3,10}^{\left(3\right)}& X_{3,13}^{\left(2\right)}& X_{3,15}^{\left(2\right)}+X_{3,11}^{\left(3\right)}& X_{3,16}^{\left(2\right)}+X_{3,12}^{\left(3\right)}& X_{3,17}^{\left(2\right)}+X_{3,13}^{\left(3\right)}\\ X_{4,10}^{\left(2\right)}+X_{48}^{\left(3\right)}& X_{4,11}^{\left(2\right)}+X_{49}^{\left(3\right)}& X_{4,12}^{\left(2\right)}+X_{4,10}^{\left(3\right)}& X_{4,13}^{\left(2\right)}& X_{4,15}^{\left(2\right)}+X_{4,11}^{\left(3\right)}& X_{4,16}^{\left(2\right)}+X_{4,12}^{\left(3\right)}& X_{4,17}^{\left(2\right)}+X_{4,13}^{\left(3\right)}\\ X_{5,10}^{\left(2\right)}+X_{58}^{\left(3\right)}& X_{5,11}^{\left(2\right)}+X_{59}^{\left(3\right)}& X_{5,12}^{\left(2\right)}+X_{5,10}^{\left(3\right)}& X_{5,13}^{\left(2\right)}& X_{5,15}^{\left(2\right)}+X_{5,11}^{\left(3\right)}& X_{5,16}^{\left(2\right)}+X_{5,12}^{\left(3\right)}& X_{5,17}^{\left(2\right)}+X_{5,13}^{\left(3\right)}\\ X_{6,10}^{\left(2\right)}& X_{6,11}^{\left(2\right)}& X_{6,12}^{\left(2\right)}& X_{6,13}^{\left(2\right)}& X_{6,15}^{\left(2\right)}& X_{6,16}^{\left(2\right)}& X_{6,17}^{\left(2\right)}\\ X_{8,10}^{\left(2\right)}+X_{68}^{\left(3\right)}& X_{8,11}^{\left(2\right)}+X_{69}^{\left(3\right)}& X_{8,12}^{\left(2\right)}+X_{6,10}^{\left(3\right)}& X_{8,13}^{\left(2\right)}& X_{8,15}^{\left(2\right)}+X_{6,11}^{\left(3\right)}& X_{8,16}^{\left(2\right)}+X_{6,12}^{\left(3\right)}& X_{8,17}^{\left(2\right)}+X_{6,13}^{\left(3\right)}\\ X_{9,10}^{\left(2\right)}+X_{78}^{\left(3\right)}& X_{9,11}^{\left(2\right)}+X_{79}^{\left(3\right)}& X_{9,12}^{\left(2\right)}+X_{7,10}^{\left(3\right)}& X_{9,13}^{\left(2\right)}& X_{9,15}^{\left(2\right)}+X_{7,11}^{\left(3\right)}& X_{9,16}^{\left(2\right)}+X_{7,12}^{\left(3\right)}& X_{9,17}^{\left(2\right)}+X_{7,13}^{\left(3\right)}\\ X_{10,10}^{\left(2\right)}+X_{88}^{\left(3\right)}& X_{10,11}^{\left(2\right)}+X_{89}^{\left(3\right)}& X_{10,12}^{\left(2\right)}+X_{8,10}^{\left(3\right)}& X_{10,13}^{\left(2\right)}& X_{10,15}^{\left(2\right)}+X_{8,11}^{\left(3\right)}& X_{10,16}^{\left(2\right)}+X_{8,12}^{\left(3\right)}& X_{10,17}^{\left(2\right)}+X_{8,13}^{\left(3\right)}\\ X_{11,10}^{\left(2\right)}+X_{98}^{\left(3\right)}& X_{11,11}^{\left(2\right)}+X_{99}^{\left(3\right)}& X_{11,12}^{\left(2\right)}+X_{9,10}^{\left(3\right)}& X_{11,13}^{\left(2\right)}& X_{11,15}^{\left(2\right)}+X_{9,11}^{\left(3\right)}& X_{11,16}^{\left(2\right)}+X_{9,12}^{\left(3\right)}& X_{11,17}^{\left(2\right)}+X_{9,13}^{\left(3\right)}\\ X_{12,10}^{\left(2\right)}+X_{10,8}^{\left(3\right)}& X_{12,11}^{\left(2\right)}+X_{10,9}^{\left(3\right)}& X_{12,12}^{\left(2\right)}+X_{10,10}^{\left(3\right)}& X_{12,13}^{\left(2\right)}& X_{12,15}^{\left(2\right)}+X_{10,11}^{\left(3\right)}& X_{12,16}^{\left(2\right)}+X_{10,12}^{\left(3\right)}& X_{12,17}^{\left(2\right)}+X_{10,13}^{\left(3\right)}\\ X_{13,10}^{\left(2\right)}& X_{13,11}^{\left(2\right)}& X_{13,12}^{\left(2\right)}& X_{13,13}^{\left(2\right)}& X_{13,15}^{\left(2\right)}& X_{13,16}^{\left(2\right)}& X_{13,17}^{\left(2\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\Phi }_{22}^{\left(2\right)}=\left(\begin{array}{ccccccc}{X}_{15,10}^{\left(2\right)}+X_{11,8}^{\left(3\right)}& X_{15,11}^{\left(2\right)}+X_{11,9}^{\left(3\right)}& X_{15,12}^{\left(2\right)}+X_{11,10}^{\left(3\right)}& X_{15,13}^{\left(2\right)}& X_{15,15}^{\left(2\right)}+X_{11,11}^{\left(3\right)}& X_{15,16}^{\left(2\right)}+X_{11,12}^{\left(3\right)}& X_{15,17}^{\left(2\right)}+X_{11,13}^{\left(3\right)}\\ X_{16,10}^{\left(2\right)}+X_{12,8}^{\left(3\right)}& X_{16,11}^{\left(2\right)}+X_{12,9}^{\left(3\right)}& X_{16,12}^{\left(2\right)}+X_{12,10}^{\left(3\right)}& X_{16,13}^{\left(2\right)}& X_{16,15}^{\left(2\right)}+X_{12,11}^{\left(3\right)}& X_{16,16}^{\left(2\right)}+X_{12,12}^{\left(3\right)}& X_{16,17}^{\left(2\right)}+X_{12,13}^{\left(3\right)}\\ X_{17,10}^{\left(2\right)}+X_{13,8}^{\left(3\right)}& X_{17,11}^{\left(2\right)}+X_{13,9}^{\left(3\right)}& X_{17,12}^{\left(2\right)}+X_{13,10}^{\left(3\right)}& X_{17,13}^{\left(2\right)}& X_{17,15}^{\left(2\right)}+X_{13,11}^{\left(3\right)}& X_{17,16}^{\left(2\right)}+X_{13,12}^{\left(3\right)}& X_{17,17}^{\left(2\right)}+X_{13,13}^{\left(3\right)}\\ X_{18,10}^{\left(2\right)}+X_{14,8}^{\left(3\right)}& X_{18,11}^{\left(2\right)}+X_{14,9}^{\left(3\right)}& X_{18,12}^{\left(2\right)}+X_{14,10}^{\left(3\right)}& X_{18,13}^{\left(2\right)}& X_{18,15}^{\left(2\right)}+X_{14,11}^{\left(3\right)}& X_{18,16}^{\left(2\right)}+X_{14,12}^{\left(3\right)}& X_{18,17}^{\left(2\right)}+X_{14,13}^{\left(3\right)}\\ X_{19,10}^{\left(2\right)}+X_{15,8}^{\left(3\right)}& X_{19,11}^{\left(2\right)}+X_{15,9}^{\left(3\right)}& X_{19,12}^{\left(2\right)}+X_{15,10}^{\left(3\right)}& X_{19,13}^{\left(2\right)}& X_{19,15}^{\left(2\right)}+X_{15,11}^{\left(3\right)}& X_{19,16}^{\left(2\right)}+X_{15,12}^{\left(3\right)}& X_{19,17}^{\left(2\right)}+X_{15,13}^{\left(3\right)}\\ X_{20,10}^{\left(2\right)}& X_{20,11}^{\left(2\right)}& X_{20,12}^{\left(2\right)}& X_{20,13}^{\left(2\right)}& X_{20,15}^{\left(2\right)}& X_{20,16}^{\left(2\right)}& X_{20,17}^{\left(2\right)}\\ X_{16,8}^{\left(3\right)}& X_{16,9}^{\left(3\right)}& X_{16,10}^{\left(3\right)}& 0& X_{16,11}^{\left(3\right)}& X_{16,12}^{\left(3\right)}& X_{16,13}^{\left(3\right)}\\ X_{17,8}^{\left(3\right)}& X_{17,9}^{\left(3\right)}& X_{17,10}^{\left(3\right)}& 0& X_{17,11}^{\left(3\right)}& X_{17,12}^{\left(3\right)}& X_{17,13}^{\left(3\right)}\\ X_{18,8}^{\left(3\right)}& X_{18,9}^{\left(3\right)}& X_{18,10}^{\left(3\right)}& 0& X_{18,11}^{\left(3\right)}& X_{18,12}^{\left(3\right)}& X_{18,13}^{\left(3\right)}\\ X_{19,8}^{\left(3\right)}& X_{19,9}^{\left(3\right)}& X_{19,10}^{\left(3\right)}& 0& X_{19,11}^{\left(3\right)}& X_{19,12}^{\left(3\right)}& X_{19,13}^{\left(3\right)}\\ X_{20,8}^{\left(3\right)}& X_{20,9}^{\left(3\right)}& X_{20,10}^{\left(3\right)}& 0& X_{20,11}^{\left(3\right)}& X_{20,12}^{\left(3\right)}& X_{20,13}^{\left(3\right)}\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\Phi }_{13}^{\left(2\right)}=\left(\begin{array}{ccccccccc}{X}_{1,18}^{\left(2\right)}+X_{1,14}^{\left(3\right)}& X_{1,19}^{\left(2\right)}+X_{1,15}^{\left(3\right)}& X_{1,20}^{\left(2\right)}& X_{1,16}^{\left(3\right)}& X_{1,17}^{\left(3\right)}& X_{1,18}^{\left(3\right)}& X_{1,19}^{\left(3\right)}& X_{1,20}^{\left(3\right)}& 0\\ X_{2,18}^{\left(2\right)}+X_{2,14}^{\left(3\right)}& X_{2,19}^{\left(2\right)}+X_{2,15}^{\left(3\right)}& X_{2,20}^{\left(2\right)}& X_{2,16}^{\left(3\right)}& X_{2,17}^{\left(3\right)}& X_{2,18}^{\left(3\right)}& X_{2,19}^{\left(3\right)}& X_{2,20}^{\left(3\right)}& 0\\ X_{3,18}^{\left(2\right)}+X_{3,14}^{\left(3\right)}& X_{3,19}^{\left(2\right)}+X_{3,15}^{\left(3\right)}& X_{3,20}^{\left(2\right)}& X_{3,16}^{\left(3\right)}& X_{3,17}^{\left(3\right)}& X_{3,18}^{\left(3\right)}& X_{3,19}^{\left(3\right)}& X_{3,20}^{\left(3\right)}& 0\\ X_{4,18}^{\left(2\right)}+X_{4,14}^{\left(3\right)}& X_{4,19}^{\left(2\right)}+X_{4,15}^{\left(3\right)}& X_{4,20}^{\left(2\right)}& X_{4,16}^{\left(3\right)}& X_{4,17}^{\left(3\right)}& X_{4,18}^{\left(3\right)}& X_{4,19}^{\left(3\right)}& X_{4,20}^{\left(3\right)}& 0\\ X_{5,18}^{\left(2\right)}+X_{5,14}^{\left(3\right)}& X_{5,19}^{\left(2\right)}+X_{5,15}^{\left(3\right)}& X_{5,20}^{\left(2\right)}& X_{5,16}^{\left(3\right)}& X_{5,17}^{\left(3\right)}& X_{5,18}^{\left(3\right)}& X_{5,19}^{\left(3\right)}& X_{5,20}^{\left(3\right)}& 0\\ X_{6,18}^{\left(2\right)}& X_{6,19}^{\left(2\right)}& X_{6,20}^{\left(2\right)}& 0& 0& 0& 0& 0& 0\\ X_{8,18}^{\left(2\right)}+X_{6,14}^{\left(3\right)}& X_{8,19}^{\left(2\right)}+X_{6,15}^{\left(3\right)}& X_{8,20}^{\left(2\right)}& X_{6,16}^{\left(3\right)}& X_{6,17}^{\left(3\right)}& X_{6,18}^{\left(3\right)}& X_{6,19}^{\left(3\right)}& X_{6,20}^{\left(3\right)}& 0\\ X_{9,18}^{\left(2\right)}+X_{7,14}^{\left(3\right)}& X_{9,19}^{\left(2\right)}+X_{7,15}^{\left(3\right)}& X_{9,20}^{\left(2\right)}& X_{7,16}^{\left(3\right)}& X_{7,17}^{\left(3\right)}& X_{7,18}^{\left(3\right)}& X_{7,19}^{\left(3\right)}& X_{7,20}^{\left(3\right)}& 0\\ X_{10,18}^{\left(2\right)}+X_{8,14}^{\left(3\right)}& X_{10,19}^{\left(2\right)}+X_{8,15}^{\left(3\right)}& X_{10,20}^{\left(2\right)}& X_{8,16}^{\left(3\right)}& X_{8,17}^{\left(3\right)}& X_{8,18}^{\left(3\right)}& X_{8,19}^{\left(3\right)}& X_{8,20}^{\left(3\right)}& 0\\ X_{11,18}^{\left(2\right)}+X_{9,14}^{\left(3\right)}& X_{11,19}^{\left(2\right)}+X_{9,15}^{\left(3\right)}& X_{11,20}^{\left(2\right)}& X_{9,16}^{\left(3\right)}& X_{9,17}^{\left(3\right)}& X_{9,18}^{\left(3\right)}& X_{9,19}^{\left(3\right)}& X_{9,20}^{\left(3\right)}& 0\\ X_{12,18}^{\left(2\right)}+X_{10,14}^{\left(3\right)}& X_{12,19}^{\left(2\right)}+X_{10,15}^{\left(3\right)}& X_{12,20}^{\left(2\right)}& X_{10,16}^{\left(3\right)}& X_{10,17}^{\left(3\right)}& X_{10,18}^{\left(3\right)}& X_{10,19}^{\left(3\right)}& X_{10,20}^{\left(3\right)}& 0\\ X_{13,18}^{\left(2\right)}& X_{13,19}^{\left(2\right)}& X_{13,20}^{\left(2\right)}& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\Phi }_{23}^{\left(2\right)}=\left(\begin{array}{ccccccccc}{X}_{15,18}^{\left(2\right)}+X_{11,14}^{\left(3\right)}& X_{15,19}^{\left(2\right)}+X_{11,15}^{\left(3\right)}& X_{15,20}^{\left(2\right)}& X_{11,16}^{\left(3\right)}& X_{11,17}^{\left(3\right)}& X_{11,18}^{\left(3\right)}& X_{11,19}^{\left(3\right)}& X_{11,20}^{\left(3\right)}& 0\\ X_{16,18}^{\left(2\right)}+X_{12,14}^{\left(3\right)}& X_{16,19}^{\left(2\right)}+X_{12,15}^{\left(3\right)}& X_{16,20}^{\left(2\right)}& X_{12,16}^{\left(3\right)}& X_{12,17}^{\left(3\right)}& X_{12,18}^{\left(3\right)}& X_{12,19}^{\left(3\right)}& X_{12,20}^{\left(3\right)}& 0\\ X_{17,18}^{\left(2\right)}+X_{13,14}^{\left(3\right)}& X_{17,19}^{\left(2\right)}+X_{13,15}^{\left(3\right)}& X_{17,20}^{\left(2\right)}& X_{13,16}^{\left(3\right)}& X_{13,17}^{\left(3\right)}& X_{13,18}^{\left(3\right)}& X_{13,19}^{\left(3\right)}& X_{13,20}^{\left(3\right)}& 0\\ X_{18,18}^{\left(2\right)}+X_{14,14}^{\left(3\right)}& X_{18,19}^{\left(2\right)}+X_{14,15}^{\left(3\right)}& X_{18,20}^{\left(2\right)}& X_{14,16}^{\left(3\right)}& X_{14,17}^{\left(3\right)}& X_{14,18}^{\left(3\right)}& X_{14,19}^{\left(3\right)}& X_{14,20}^{\left(3\right)}& 0\\ X_{19,18}^{\left(2\right)}+X_{15,14}^{\left(3\right)}& X_{19,19}^{\left(2\right)}+X_{15,15}^{\left(3\right)}& X_{19,20}^{\left(2\right)}& X_{15,16}^{\left(3\right)}& X_{15,17}^{\left(3\right)}& X_{15,18}^{\left(3\right)}& X_{15,19}^{\left(3\right)}& X_{15,20}^{\left(3\right)}& 0\\ X_{20,18}^{\left(2\right)}& X_{20,19}^{\left(2\right)}& X_{20,20}^{\left(2\right)}& 0& 0& 0& 0& 0& 0\\ X_{16,14}^{\left(3\right)}& X_{16,15}^{\left(3\right)}& 0& X_{16,16}^{\left(3\right)}& X_{16,17}^{\left(3\right)}& X_{16,18}^{\left(3\right)}& X_{16,19}^{\left(3\right)}& X_{16,20}^{\left(3\right)}& 0\\ X_{17,14}^{\left(3\right)}& X_{17,15}^{\left(3\right)}& 0& X_{17,16}^{\left(3\right)}& X_{17,17}^{\left(3\right)}& X_{17,18}^{\left(3\right)}& X_{17,19}^{\left(3\right)}& X_{17,20}^{\left(3\right)}& 0\\ X_{18,14}^{\left(3\right)}& X_{18,15}^{\left(3\right)}& 0& X_{18,16}^{\left(3\right)}& X_{18,17}^{\left(3\right)}& X_{18,18}^{\left(3\right)}& X_{18,19}^{\left(3\right)}& X_{18,20}^{\left(3\right)}& 0\\ X_{19,14}^{\left(3\right)}& X_{19,15}^{\left(3\right)}& 0& X_{19,16}^{\left(3\right)}& X_{19,17}^{\left(3\right)}& X_{19,18}^{\left(3\right)}& X_{19,19}^{\left(3\right)}& X_{19,20}^{\left(3\right)}& 0\\ X_{20,14}^{\left(3\right)}& X_{20,15}^{\left(3\right)}& 0& X_{20,16}^{\left(3\right)}& X_{20,17}^{\left(3\right)}& X_{20,18}^{\left(3\right)}& X_{20,19}^{\left(3\right)}& X_{20,20}^{\left(3\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right).\end{array}$$

We have

$$\begin{array}{c}\hfill \widehat{{\Omega }_3}=\left(\begin{array}{ccc}{\Phi }_{11}^{\left(3\right)}& {\Phi }_{12}^{\left(3\right)}& {\Phi }_{13}^{\left(3\right)}\\ {\Phi }_{21}^{\left(3\right)}& {\Phi }_{22}^{\left(3\right)}& {\Phi }_{23}^{\left(3\right)}\end{array}\right),\end{array}$$

(10)

where

$$\begin{array}{c}\hfill {\Phi }_{11}^{\left(3\right)}=\left(\begin{array}{cccccccc}{X}_{11}^{\left(3\right)}+X_{11}^{\left(4\right)}& X_{12}^{\left(3\right)}+X_{12}^{\left(4\right)}& X_{13}^{\left(3\right)}+X_{13}^{\left(4\right)}& X_{14}^{\left(3\right)}& X_{16}^{\left(3\right)}+X_{14}^{\left(4\right)}& X_{17}^{\left(3\right)}+X_{15}^{\left(4\right)}& X_{18}^{\left(3\right)}+X_{16}^{\left(4\right)}& X_{19}^{\left(3\right)}\\ X_{21}^{\left(3\right)}+X_{21}^{\left(4\right)}& X_{22}^{\left(3\right)}+X_{22}^{\left(4\right)}& X_{23}^{\left(3\right)}+X_{23}^{\left(4\right)}& X_{24}^{\left(3\right)}& X_{26}^{\left(3\right)}+X_{24}^{\left(4\right)}& X_{27}^{\left(3\right)}+X_{25}^{\left(4\right)}& X_{28}^{\left(3\right)}+X_{26}^{\left(4\right)}& X_{29}^{\left(3\right)}\\ X_{31}^{\left(3\right)}+X_{31}^{\left(4\right)}& X_{32}^{\left(3\right)}+X_{32}^{\left(4\right)}& X_{33}^{\left(3\right)}+X_{33}^{\left(4\right)}& X_{34}^{\left(3\right)}& X_{36}^{\left(3\right)}+X_{34}^{\left(4\right)}& X_{37}^{\left(3\right)}+X_{35}^{\left(4\right)}& X_{38}^{\left(3\right)}+X_{36}^{\left(4\right)}& X_{39}^{\left(3\right)}\\ X_{41}^{\left(3\right)}& X_{42}^{\left(3\right)}& X_{43}^{\left(3\right)}& X_{44}^{\left(3\right)}& X_{46}^{\left(3\right)}& X_{47}^{\left(3\right)}& X_{48}^{\left(3\right)}& X_{49}^{\left(3\right)}\\ X_{61}^{\left(3\right)}+X_{41}^{\left(4\right)}& X_{62}^{\left(3\right)}+X_{42}^{\left(4\right)}& X_{63}^{\left(3\right)}+X_{43}^{\left(4\right)}& X_{64}^{\left(3\right)}& X_{66}^{\left(3\right)}+X_{44}^{\left(4\right)}& X_{67}^{\left(3\right)}+X_{45}^{\left(4\right)}& X_{68}^{\left(3\right)}+X_{46}^{\left(4\right)}& X_{69}^{\left(3\right)}\\ X_{71}^{\left(3\right)}+X_{51}^{\left(4\right)}& X_{72}^{\left(3\right)}+X_{52}^{\left(4\right)}& X_{73}^{\left(3\right)}+X_{53}^{\left(4\right)}& X_{74}^{\left(3\right)}& X_{76}^{\left(3\right)}+X_{54}^{\left(4\right)}& X_{77}^{\left(3\right)}+X_{55}^{\left(4\right)}& X_{78}^{\left(3\right)}+X_{56}^{\left(4\right)}& X_{79}^{\left(3\right)}\\ X_{81}^{\left(3\right)}+X_{61}^{\left(4\right)}& X_{82}^{\left(3\right)}+X_{62}^{\left(4\right)}& X_{83}^{\left(3\right)}+X_{63}^{\left(4\right)}& X_{84}^{\left(3\right)}& X_{86}^{\left(3\right)}+X_{64}^{\left(4\right)}& X_{87}^{\left(3\right)}+X_{65}^{\left(4\right)}& X_{88}^{\left(3\right)}+X_{66}^{\left(4\right)}& X_{89}^{\left(3\right)}\\ X_{91}^{\left(3\right)}& X_{92}^{\left(3\right)}& X_{93}^{\left(3\right)}& X_{94}^{\left(3\right)}& X_{96}^{\left(3\right)}& X_{97}^{\left(3\right)}& X_{98}^{\left(3\right)}& X_{99}^{\left(3\right)}\\ X_{11,1}^{\left(3\right)}+X_{71}^{\left(4\right)}& X_{11,2}^{\left(3\right)}+X_{72}^{\left(4\right)}& X_{11,3}^{\left(3\right)}+X_{73}^{\left(4\right)}& X_{11,4}^{\left(3\right)}& X_{11,6}^{\left(3\right)}+X_{74}^{\left(4\right)}& X_{11,7}^{\left(3\right)}+X_{75}^{\left(4\right)}& X_{11,8}^{\left(3\right)}+X_{76}^{\left(4\right)}& X_{11,9}^{\left(3\right)}\\ X_{12,1}^{\left(3\right)}+X_{81}^{\left(4\right)}& X_{12,2}^{\left(3\right)}+X_{82}^{\left(4\right)}& X_{12,3}^{\left(3\right)}+X_{83}^{\left(4\right)}& X_{12,4}^{\left(3\right)}& X_{12,6}^{\left(3\right)}+X_{84}^{\left(4\right)}& X_{12,7}^{\left(3\right)}+X_{85}^{\left(4\right)}& X_{12,8}^{\left(3\right)}+X_{86}^{\left(4\right)}& X_{12,9}^{\left(3\right)}\\ X_{13,1}^{\left(3\right)}+X_{91}^{\left(4\right)}& X_{13,2}^{\left(3\right)}+X_{92}^{\left(4\right)}& X_{13,3}^{\left(3\right)}+X_{93}^{\left(4\right)}& X_{13,4}^{\left(3\right)}& X_{13,6}^{\left(3\right)}+X_{94}^{\left(4\right)}& X_{13,7}^{\left(3\right)}+X_{95}^{\left(4\right)}& X_{13,8}^{\left(3\right)}+X_{96}^{\left(4\right)}& X_{13,9}^{\left(3\right)}\\ X_{14,1}^{\left(3\right)}& X_{14,2}^{\left(3\right)}& X_{14,3}^{\left(3\right)}& X_{14,4}^{\left(3\right)}& X_{14,6}^{\left(3\right)}& X_{14,7}^{\left(3\right)}& X_{14,8}^{\left(3\right)}& X_{14,9}^{\left(3\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\Phi }_{21}^{\left(3\right)}=\left(\begin{array}{cccccccc}{X}_{16,1}^{\left(3\right)}+X_{10,1}^{\left(4\right)}& X_{16,2}^{\left(3\right)}+X_{10,2}^{\left(4\right)}& X_{16,3}^{\left(3\right)}+X_{10,3}^{\left(4\right)}& X_{16,4}^{\left(3\right)}& X_{16,6}^{\left(3\right)}+X_{10,4}^{\left(4\right)}& X_{16,7}^{\left(3\right)}+X_{10,5}^{\left(4\right)}& X_{16,8}^{\left(3\right)}+X_{10,6}^{\left(4\right)}& X_{16,9}^{\left(3\right)}\\ X_{17,1}^{\left(3\right)}+X_{11,1}^{\left(4\right)}& X_{17,2}^{\left(3\right)}+X_{11,2}^{\left(4\right)}& X_{17,3}^{\left(3\right)}+X_{11,3}^{\left(4\right)}& X_{17,4}^{\left(3\right)}& X_{17,6}^{\left(3\right)}+X_{11,4}^{\left(4\right)}& X_{17,7}^{\left(3\right)}+X_{11,5}^{\left(4\right)}& X_{17,8}^{\left(3\right)}+X_{11,6}^{\left(4\right)}& X_{17,9}^{\left(3\right)}\\ X_{18,1}^{\left(3\right)}+X_{12,1}^{\left(4\right)}& X_{18,2}^{\left(3\right)}+X_{12,2}^{\left(4\right)}& X_{18,3}^{\left(3\right)}+X_{12,3}^{\left(4\right)}& X_{18,4}^{\left(3\right)}& X_{18,6}^{\left(3\right)}+X_{12,4}^{\left(4\right)}& X_{18,7}^{\left(3\right)}+X_{12,5}^{\left(4\right)}& X_{18,8}^{\left(3\right)}+X_{12,6}^{\left(4\right)}& X_{18,9}^{\left(3\right)}\\ X_{19,1}^{\left(3\right)}& X_{19,2}^{\left(3\right)}& X_{19,3}^{\left(3\right)}& X_{19,4}^{\left(3\right)}& X_{19,6}^{\left(3\right)}& X_{19,7}^{\left(3\right)}& X_{19,8}^{\left(3\right)}& X_{19,9}^{\left(3\right)}\\ X_{21,1}^{\left(3\right)}+X_{13,1}^{\left(4\right)}& X_{21,2}^{\left(3\right)}+X_{13,2}^{\left(4\right)}& X_{21,3}^{\left(3\right)}+X_{13,3}^{\left(4\right)}& X_{21,4}^{\left(3\right)}& X_{21,6}^{\left(3\right)}+X_{13,4}^{\left(4\right)}& X_{21,7}^{\left(3\right)}+X_{13,5}^{\left(4\right)}& X_{21,8}^{\left(3\right)}+X_{13,6}^{\left(4\right)}& X_{21,9}^{\left(3\right)}\\ X_{22,1}^{\left(3\right)}+X_{14,1}^{\left(4\right)}& X_{22,2}^{\left(3\right)}+X_{14,2}^{\left(4\right)}& X_{22,3}^{\left(3\right)}+X_{14,3}^{\left(4\right)}& X_{22,4}^{\left(3\right)}& X_{22,6}^{\left(3\right)}+X_{14,4}^{\left(4\right)}& X_{22,7}^{\left(3\right)}+X_{14,5}^{\left(4\right)}& X_{22,8}^{\left(3\right)}+X_{14,6}^{\left(4\right)}& X_{22,9}^{\left(3\right)}\\ X_{23,1}^{\left(3\right)}+X_{15,1}^{\left(4\right)}& X_{23,2}^{\left(3\right)}+X_{15,2}^{\left(4\right)}& X_{23,3}^{\left(3\right)}+X_{15,3}^{\left(4\right)}& X_{23,4}^{\left(3\right)}& X_{23,6}^{\left(3\right)}+X_{15,4}^{\left(4\right)}& X_{23,7}^{\left(3\right)}+X_{15,5}^{\left(4\right)}& X_{23,8}^{\left(3\right)}+X_{15,6}^{\left(4\right)}& X_{23,9}^{\left(3\right)}\\ X_{24,1}^{\left(3\right)}& X_{24,2}^{\left(3\right)}& X_{24,3}^{\left(3\right)}& X_{24,4}^{\left(3\right)}& X_{24,6}^{\left(3\right)}& X_{24,7}^{\left(3\right)}& X_{24,8}^{\left(3\right)}& X_{24,9}^{\left(3\right)}\\ X_{16,1}^{\left(4\right)}& X_{16,2}^{\left(4\right)}& X_{16,3}^{\left(4\right)}& 0& X_{16,4}^{\left(4\right)}& X_{16,5}^{\left(4\right)}& X_{16,6}^{\left(4\right)}& 0\\ X_{17,1}^{\left(4\right)}& X_{17,2}^{\left(4\right)}& X_{17,3}^{\left(4\right)}& 0& X_{17,4}^{\left(4\right)}& X_{17,5}^{\left(4\right)}& X_{17,6}^{\left(4\right)}& 0\\ X_{18,1}^{\left(4\right)}& X_{18,2}^{\left(4\right)}& X_{18,3}^{\left(4\right)}& 0& X_{18,4}^{\left(4\right)}& X_{18,5}^{\left(4\right)}& X_{18,6}^{\left(4\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{cc}\hfill {\Phi }_{12}^{\left(3\right)}=& \left(\begin{array}{cccccccc}{X}_{1,11}^{\left(3\right)}+X_{17}^{\left(4\right)}& X_{1,12}^{\left(3\right)}+X_{18}^{\left(4\right)}& X_{1,13}^{\left(3\right)}+X_{19}^{\left(4\right)}& X_{1,14}^{\left(3\right)}& X_{1,16}^{\left(3\right)}+X_{1,10}^{\left(4\right)}& X_{1,17}^{\left(3\right)}+X_{1,11}^{\left(4\right)}& X_{1,18}^{\left(3\right)}+X_{1,12}^{\left(4\right)}& X_{1,19}^{\left(3\right)}\\ X_{2,11}^{\left(3\right)}+X_{27}^{\left(4\right)}& X_{2,12}^{\left(3\right)}+X_{28}^{\left(4\right)}& X_{2,13}^{\left(3\right)}+X_{29}^{\left(4\right)}& X_{2,14}^{\left(3\right)}& X_{2,16}^{\left(3\right)}+X_{2,10}^{\left(4\right)}& X_{2,17}^{\left(3\right)}+X_{2,11}^{\left(4\right)}& X_{2,18}^{\left(3\right)}+X_{2,12}^{\left(4\right)}& X_{2,19}^{\left(3\right)}\\ X_{3,11}^{\left(3\right)}+X_{37}^{\left(4\right)}& X_{3,12}^{\left(3\right)}+X_{38}^{\left(4\right)}& X_{3,13}^{\left(3\right)}+X_{39}^{\left(4\right)}& X_{3,14}^{\left(3\right)}& X_{3,16}^{\left(3\right)}+X_{3,10}^{\left(4\right)}& X_{3,17}^{\left(3\right)}+X_{3,11}^{\left(4\right)}& X_{3,18}^{\left(3\right)}+X_{3,12}^{\left(4\right)}& X_{3,19}^{\left(3\right)}\\ X_{4,11}^{\left(3\right)}& X_{4,12}^{\left(3\right)}& X_{4,13}^{\left(3\right)}& X_{4,14}^{\left(3\right)}& X_{4,16}^{\left(3\right)}& X_{4,17}^{\left(3\right)}& X_{4,18}^{\left(3\right)}& X_{4,19}^{\left(3\right)}\\ X_{6,11}^{\left(3\right)}+X_{47}^{\left(4\right)}& X_{6,12}^{\left(3\right)}+X_{48}^{\left(4\right)}& X_{6,13}^{\left(3\right)}+X_{49}^{\left(4\right)}& X_{6,14}^{\left(3\right)}& X_{6,16}^{\left(3\right)}+X_{4,10}^{\left(4\right)}& X_{6,17}^{\left(3\right)}+X_{4,11}^{\left(4\right)}& X_{6,18}^{\left(3\right)}+X_{4,12}^{\left(4\right)}& X_{6,19}^{\left(3\right)}\\ X_{7,11}^{\left(3\right)}+X_{57}^{\left(4\right)}& X_{7,12}^{\left(3\right)}+X_{58}^{\left(4\right)}& X_{7,13}^{\left(3\right)}+X_{59}^{\left(4\right)}& X_{7,14}^{\left(3\right)}& X_{7,16}^{\left(3\right)}+X_{5,10}^{\left(4\right)}& X_{7,17}^{\left(3\right)}+X_{5,11}^{\left(4\right)}& X_{7,18}^{\left(3\right)}+X_{5,12}^{\left(4\right)}& X_{7,19}^{\left(3\right)}\\ X_{8,11}^{\left(3\right)}+X_{67}^{\left(4\right)}& X_{8,12}^{\left(3\right)}+X_{68}^{\left(4\right)}& X_{8,13}^{\left(3\right)}+X_{69}^{\left(4\right)}& X_{8,14}^{\left(3\right)}& X_{8,16}^{\left(3\right)}+X_{6,10}^{\left(4\right)}& X_{8,17}^{\left(3\right)}+X_{6,11}^{\left(4\right)}& X_{8,18}^{\left(3\right)}+X_{6,12}^{\left(4\right)}& X_{8,19}^{\left(3\right)}\\ X_{9,11}^{\left(3\right)}& X_{9,12}^{\left(3\right)}& X_{9,13}^{\left(3\right)}& X_{9,14}^{\left(3\right)}& X_{9,16}^{\left(3\right)}& X_{9,17}^{\left(3\right)}& X_{9,18}^{\left(3\right)}& X_{9,19}^{\left(3\right)}\\ X_{11,11}^{\left(3\right)}+X_{77}^{\left(4\right)}& X_{11,12}^{\left(3\right)}+X_{78}^{\left(4\right)}& X_{11,13}^{\left(3\right)}+X_{79}^{\left(4\right)}& X_{11,14}^{\left(3\right)}& X_{11,16}^{\left(3\right)}+X_{7,10}^{\left(4\right)}& X_{11,17}^{\left(3\right)}+X_{7,11}^{\left(4\right)}& X_{11,18}^{\left(3\right)}+X_{7,12}^{\left(4\right)}& X_{11,19}^{\left(3\right)}\\ X_{12,11}^{\left(3\right)}+X_{87}^{\left(4\right)}& X_{12,12}^{\left(3\right)}+X_{88}^{\left(4\right)}& X_{12,13}^{\left(3\right)}+X_{89}^{\left(4\right)}& X_{12,14}^{\left(3\right)}& X_{12,16}^{\left(3\right)}+X_{8,10}^{\left(4\right)}& X_{12,17}^{\left(3\right)}+X_{8,11}^{\left(4\right)}& X_{12,18}^{\left(3\right)}+X_{8,12}^{\left(4\right)}& X_{12,19}^{\left(3\right)}\\ X_{13,11}^{\left(3\right)}+X_{97}^{\left(4\right)}& X_{13,12}^{\left(3\right)}+X_{98}^{\left(4\right)}& X_{13,13}^{\left(3\right)}+X_{99}^{\left(4\right)}& X_{13,14}^{\left(3\right)}& X_{13,16}^{\left(3\right)}+X_{9,10}^{\left(4\right)}& X_{13,17}^{\left(3\right)}+X_{9,11}^{\left(4\right)}& X_{13,18}^{\left(3\right)}+X_{9,12}^{\left(4\right)}& X_{13,19}^{\left(3\right)}\\ X_{14,11}^{\left(3\right)}& X_{14,12}^{\left(3\right)}& X_{14,13}^{\left(3\right)}& X_{14,14}^{\left(3\right)}& X_{14,16}^{\left(3\right)}& X_{14,17}^{\left(3\right)}& X_{14,18}^{\left(3\right)}& X_{14,19}^{\left(3\right)}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\Phi }_{22}^{\left(3\right)}=& \left(\begin{array}{cccccccc}{X}_{16,11}^{\left(3\right)}+X_{10,7}^{\left(4\right)}& X_{16,12}^{\left(3\right)}+X_{10,8}^{\left(4\right)}& X_{16,13}^{\left(3\right)}+X_{10,9}^{\left(4\right)}& X_{16,14}^{\left(3\right)}& X_{16,16}^{\left(3\right)}+X_{10,10}^{\left(4\right)}& X_{16,17}^{\left(3\right)}+X_{10,11}^{\left(4\right)}& X_{16,18}^{\left(3\right)}+X_{10,12}^{\left(4\right)}& X_{16,19}^{\left(3\right)}\\ X_{17,11}^{\left(3\right)}+X_{11,7}^{\left(4\right)}& X_{17,12}^{\left(3\right)}+X_{11,8}^{\left(4\right)}& X_{17,13}^{\left(3\right)}+X_{11,9}^{\left(4\right)}& X_{17,14}^{\left(3\right)}& X_{17,16}^{\left(3\right)}+X_{11,10}^{\left(4\right)}& X_{17,17}^{\left(3\right)}+X_{11,11}^{\left(4\right)}& X_{17,18}^{\left(3\right)}+X_{11,12}^{\left(4\right)}& X_{17,19}^{\left(3\right)}\\ X_{18,11}^{\left(3\right)}+X_{12,7}^{\left(4\right)}& X_{18,12}^{\left(3\right)}+X_{12,8}^{\left(4\right)}& X_{18,13}^{\left(3\right)}+X_{12,9}^{\left(4\right)}& X_{18,14}^{\left(3\right)}& X_{18,16}^{\left(3\right)}+X_{12,10}^{\left(4\right)}& X_{18,17}^{\left(3\right)}+X_{12,11}^{\left(4\right)}& X_{18,18}^{\left(3\right)}+X_{12,12}^{\left(4\right)}& X_{18,19}^{\left(3\right)}\\ X_{19,11}^{\left(3\right)}& X_{19,12}^{\left(3\right)}& X_{19,13}^{\left(3\right)}& X_{19,14}^{\left(3\right)}& X_{19,16}^{\left(3\right)}& X_{19,17}^{\left(3\right)}& X_{19,18}^{\left(3\right)}& X_{19,19}^{\left(3\right)}\\ X_{21,11}^{\left(3\right)}+X_{13,7}^{\left(4\right)}& X_{21,12}^{\left(3\right)}+X_{13,8}^{\left(4\right)}& X_{21,13}^{\left(3\right)}+X_{13,9}^{\left(4\right)}& X_{21,14}^{\left(3\right)}& X_{21,16}^{\left(3\right)}+X_{13,10}^{\left(4\right)}& X_{21,17}^{\left(3\right)}+X_{13,11}^{\left(4\right)}& X_{21,18}^{\left(3\right)}+X_{13,12}^{\left(4\right)}& X_{21,19}^{\left(3\right)}\\ X_{22,11}^{\left(3\right)}+X_{14,7}^{\left(4\right)}& X_{22,12}^{\left(3\right)}+X_{14,8}^{\left(4\right)}& X_{22,13}^{\left(3\right)}+X_{14,9}^{\left(4\right)}& X_{22,14}^{\left(3\right)}& X_{22,16}^{\left(3\right)}+X_{14,10}^{\left(4\right)}& X_{22,17}^{\left(3\right)}+X_{14,11}^{\left(4\right)}& X_{22,18}^{\left(3\right)}+X_{14,12}^{\left(4\right)}& X_{22,19}^{\left(3\right)}\\ X_{23,11}^{\left(3\right)}+X_{15,7}^{\left(4\right)}& X_{23,12}^{\left(3\right)}+X_{15,8}^{\left(4\right)}& X_{23,13}^{\left(3\right)}+X_{15,9}^{\left(4\right)}& X_{23,14}^{\left(3\right)}& X_{23,16}^{\left(3\right)}+X_{15,10}^{\left(4\right)}& X_{23,17}^{\left(3\right)}+X_{15,11}^{\left(4\right)}& X_{23,18}^{\left(3\right)}+X_{15,12}^{\left(4\right)}& X_{23,19}^{\left(3\right)}\\ X_{24,11}^{\left(3\right)}& X_{24,12}^{\left(3\right)}& X_{24,13}^{\left(3\right)}& X_{24,14}^{\left(3\right)}& X_{24,16}^{\left(3\right)}& X_{24,17}^{\left(3\right)}& X_{24,18}^{\left(3\right)}& X_{24,19}^{\left(3\right)}\\ X_{16,7}^{\left(4\right)}& X_{16,8}^{\left(4\right)}& X_{16,9}^{\left(4\right)}& 0& X_{16,10}^{\left(4\right)}& X_{16,11}^{\left(4\right)}& X_{16,12}^{\left(4\right)}& 0\\ X_{17,7}^{\left(4\right)}& X_{17,8}^{\left(4\right)}& X_{17,9}^{\left(4\right)}& 0& X_{17,10}^{\left(4\right)}& X_{17,11}^{\left(4\right)}& X_{17,12}^{\left(4\right)}& 0\\ X_{18,7}^{\left(4\right)}& X_{18,8}^{\left(4\right)}& X_{18,9}^{\left(4\right)}& 0& X_{18,10}^{\left(4\right)}& X_{18,11}^{\left(4\right)}& X_{18,12}^{\left(4\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill {\Phi }_{13}^{\left(3\right)}=\left(\begin{array}{cccccccc}{X}_{1,21}^{\left(3\right)}+X_{1,13}^{\left(4\right)}& X_{1,22}^{\left(3\right)}+X_{1,14}^{\left(4\right)}& X_{1,23}^{\left(3\right)}+X_{1,15}^{\left(4\right)}& X_{1,24}^{\left(3\right)}& X_{1,16}^{\left(4\right)}& X_{1,17}^{\left(4\right)}& X_{1,18}^{\left(4\right)}& 0\\ X_{2,21}^{\left(3\right)}+X_{2,13}^{\left(4\right)}& X_{2,22}^{\left(3\right)}+X_{2,14}^{\left(4\right)}& X_{2,23}^{\left(3\right)}+X_{2,15}^{\left(4\right)}& X_{2,24}^{\left(3\right)}& X_{2,16}^{\left(4\right)}& X_{2,17}^{\left(4\right)}& X_{2,18}^{\left(4\right)}& 0\\ X_{3,21}^{\left(3\right)}+X_{3,13}^{\left(4\right)}& X_{3,22}^{\left(3\right)}+X_{3,14}^{\left(4\right)}& X_{3,23}^{\left(3\right)}+X_{3,15}^{\left(4\right)}& X_{3,24}^{\left(3\right)}& X_{3,16}^{\left(4\right)}& X_{3,17}^{\left(4\right)}& X_{3,18}^{\left(4\right)}& 0\\ X_{4,21}^{\left(3\right)}& X_{4,22}^{\left(3\right)}& X_{4,23}^{\left(3\right)}& X_{4,24}^{\left(3\right)}& 0& 0& 0& 0\\ X_{6,21}^{\left(3\right)}+X_{4,13}^{\left(4\right)}& X_{6,22}^{\left(3\right)}+X_{4,14}^{\left(4\right)}& X_{6,23}^{\left(3\right)}+X_{4,15}^{\left(4\right)}& X_{6,24}^{\left(3\right)}& X_{4,16}^{\left(4\right)}& X_{4,17}^{\left(4\right)}& X_{4,18}^{\left(4\right)}& 0\\ X_{7,21}^{\left(3\right)}+X_{5,13}^{\left(4\right)}& X_{7,22}^{\left(3\right)}+X_{5,14}^{\left(4\right)}& X_{7,23}^{\left(3\right)}+X_{5,15}^{\left(4\right)}& X_{7,24}^{\left(3\right)}& X_{5,16}^{\left(4\right)}& X_{5,17}^{\left(4\right)}& X_{5,18}^{\left(4\right)}& 0\\ X_{8,21}^{\left(3\right)}+X_{6,13}^{\left(4\right)}& X_{8,22}^{\left(3\right)}+X_{6,14}^{\left(4\right)}& X_{8,23}^{\left(3\right)}+X_{6,15}^{\left(4\right)}& X_{8,24}^{\left(3\right)}& X_{6,16}^{\left(4\right)}& X_{6,17}^{\left(4\right)}& X_{6,18}^{\left(4\right)}& 0\\ X_{9,21}^{\left(3\right)}& X_{9,22}^{\left(3\right)}& X_{9,23}^{\left(3\right)}& X_{9,24}^{\left(3\right)}& 0& 0& 0& 0\\ X_{11,21}^{\left(3\right)}+X_{7,13}^{\left(4\right)}& X_{11,22}^{\left(3\right)}+X_{7,14}^{\left(4\right)}& X_{11,23}^{\left(3\right)}+X_{7,15}^{\left(4\right)}& X_{11,24}^{\left(3\right)}& X_{7,16}^{\left(4\right)}& X_{7,17}^{\left(4\right)}& X_{7,18}^{\left(4\right)}& 0\\ X_{12,21}^{\left(3\right)}+X_{8,13}^{\left(4\right)}& X_{12,22}^{\left(3\right)}+X_{8,14}^{\left(4\right)}& X_{12,23}^{\left(3\right)}+X_{8,15}^{\left(4\right)}& X_{12,24}^{\left(3\right)}& X_{8,16}^{\left(4\right)}& X_{8,17}^{\left(4\right)}& X_{8,18}^{\left(4\right)}& 0\\ X_{13,21}^{\left(3\right)}+X_{9,13}^{\left(4\right)}& X_{13,22}^{\left(3\right)}+X_{9,14}^{\left(4\right)}& X_{13,23}^{\left(3\right)}+X_{9,15}^{\left(4\right)}& X_{13,24}^{\left(3\right)}& X_{9,16}^{\left(4\right)}& X_{9,17}^{\left(4\right)}& X_{9,18}^{\left(4\right)}& 0\\ X_{14,21}^{\left(3\right)}& X_{14,22}^{\left(3\right)}& X_{14,23}^{\left(3\right)}& X_{14,24}^{\left(3\right)}& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\Phi }_{23}^{\left(3\right)}=\left(\begin{array}{cccccccc}{X}_{16,21}^{\left(3\right)}+X_{10,13}^{\left(4\right)}& X_{16,22}^{\left(3\right)}+X_{10,14}^{\left(4\right)}& X_{16,23}^{\left(3\right)}+X_{10,15}^{\left(4\right)}& X_{16,24}^{\left(3\right)}& X_{10,16}^{\left(4\right)}& X_{10,17}^{\left(4\right)}& X_{10,18}^{\left(4\right)}& 0\\ X_{17,21}^{\left(3\right)}+X_{11,13}^{\left(4\right)}& X_{17,22}^{\left(3\right)}+X_{11,14}^{\left(4\right)}& X_{17,23}^{\left(3\right)}+X_{11,15}^{\left(4\right)}& X_{17,24}^{\left(3\right)}& X_{11,16}^{\left(4\right)}& X_{11,17}^{\left(4\right)}& X_{11,18}^{\left(4\right)}& 0\\ X_{18,21}^{\left(3\right)}+X_{12,13}^{\left(4\right)}& X_{18,22}^{\left(3\right)}+X_{12,14}^{\left(4\right)}& X_{18,23}^{\left(3\right)}+X_{12,15}^{\left(4\right)}& X_{18,24}^{\left(3\right)}& X_{12,16}^{\left(4\right)}& X_{12,17}^{\left(4\right)}& X_{12,18}^{\left(4\right)}& 0\\ X_{19,21}^{\left(3\right)}& X_{19,22}^{\left(3\right)}& X_{19,23}^{\left(3\right)}& X_{19,24}^{\left(3\right)}& 0& 0& 0& 0\\ X_{21,21}^{\left(3\right)}+X_{13,13}^{\left(4\right)}& X_{21,22}^{\left(3\right)}+X_{13,14}^{\left(4\right)}& X_{21,23}^{\left(3\right)}+X_{13,15}^{\left(4\right)}& X_{21,24}^{\left(3\right)}& X_{13,16}^{\left(4\right)}& X_{13,17}^{\left(4\right)}& X_{13,18}^{\left(4\right)}& 0\\ X_{22,21}^{\left(3\right)}+X_{14,13}^{\left(4\right)}& X_{22,22}^{\left(3\right)}+X_{14,14}^{\left(4\right)}& X_{22,23}^{\left(3\right)}+X_{14,15}^{\left(4\right)}& X_{22,24}^{\left(3\right)}& X_{14,16}^{\left(4\right)}& X_{14,17}^{\left(4\right)}& X_{14,18}^{\left(4\right)}& 0\\ X_{23,21}^{\left(3\right)}+X_{15,13}^{\left(4\right)}& X_{23,22}^{\left(3\right)}+X_{15,14}^{\left(4\right)}& X_{23,23}^{\left(3\right)}+X_{15,15}^{\left(4\right)}& X_{23,24}^{\left(3\right)}& X_{15,16}^{\left(4\right)}& X_{15,17}^{\left(4\right)}& X_{15,18}^{\left(4\right)}& 0\\ X_{24,21}^{\left(3\right)}& X_{24,22}^{\left(3\right)}& X_{24,23}^{\left(3\right)}& X_{24,24}^{\left(3\right)}& 0& 0& 0& 0\\ X_{16,13}^{\left(4\right)}& X_{16,14}^{\left(4\right)}& X_{16,15}^{\left(4\right)}& 0& X_{16,16}^{\left(4\right)}& X_{16,17}^{\left(4\right)}& X_{16,18}^{\left(4\right)}& 0\\ X_{17,13}^{\left(4\right)}& X_{17,14}^{\left(4\right)}& X_{17,15}^{\left(4\right)}& 0& X_{17,16}^{\left(4\right)}& X_{17,17}^{\left(4\right)}& X_{17,18}^{\left(4\right)}& 0\\ X_{18,13}^{\left(4\right)}& X_{18,14}^{\left(4\right)}& X_{18,15}^{\left(4\right)}& 0& X_{18,16}^{\left(4\right)}& X_{18,17}^{\left(4\right)}& X_{18,18}^{\left(4\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right).\end{array}$$

We have

$$\begin{array}{c}\hfill \widehat{{\Omega }_4}=({\Phi }_1^{\left(4\right)},{\Phi }_2^{\left(4\right)}),\end{array}$$

(11)

where

$$\begin{array}{c}\hfill {\Phi }_1^{\left(4\right)}=\left(\begin{array}{ccccccc}{X}_{11}^{\left(4\right)}+X_{11}^{\left(5\right)}& X_{12}^{\left(4\right)}& X_{14}^{\left(4\right)}+X_{12}^{\left(5\right)}& X_{15}^{\left(4\right)}& X_{17}^{\left(4\right)}+X_{13}^{\left(5\right)}& X_{18}^{\left(4\right)}& X_{1,10}^{\left(4\right)}+X_{14}^{\left(5\right)}\\ X_{21}^{\left(4\right)}& X_{22}^{\left(4\right)}& X_{24}^{\left(4\right)}& X_{25}^{\left(4\right)}& X_{27}^{\left(4\right)}& X_{28}^{\left(4\right)}& X_{2,10}^{\left(4\right)}\\ X_{41}^{\left(4\right)}+X_{21}^{\left(5\right)}& X_{42}^{\left(4\right)}& X_{44}^{\left(4\right)}+X_{22}^{\left(5\right)}& X_{45}^{\left(4\right)}& X_{47}^{\left(4\right)}+X_{23}^{\left(5\right)}& X_{48}^{\left(4\right)}& X_{4,10}^{\left(4\right)}+X_{24}^{\left(5\right)}\\ X_{51}^{\left(4\right)}& X_{52}^{\left(4\right)}& X_{54}^{\left(4\right)}& X_{55}^{\left(4\right)}& X_{57}^{\left(4\right)}& X_{58}^{\left(4\right)}& X_{5,10}^{\left(4\right)}\\ X_{71}^{\left(4\right)}+X_{31}^{\left(5\right)}& X_{72}^{\left(4\right)}& X_{74}^{\left(4\right)}+X_{32}^{\left(5\right)}& X_{75}^{\left(4\right)}& X_{77}^{\left(4\right)}+X_{33}^{\left(5\right)}& X_{78}^{\left(4\right)}& X_{7,10}^{\left(4\right)}+X_{34}^{\left(5\right)}\\ X_{81}^{\left(4\right)}& X_{82}^{\left(4\right)}& X_{84}^{\left(4\right)}& X_{85}^{\left(4\right)}& X_{87}^{\left(4\right)}& X_{88}^{\left(4\right)}& X_{8,10}^{\left(4\right)}\\ X_{10,1}^{\left(4\right)}+X_{41}^{\left(5\right)}& X_{10,2}^{\left(4\right)}& X_{10,4}^{\left(4\right)}+X_{42}^{\left(5\right)}& X_{10,5}^{\left(4\right)}& X_{10,7}^{\left(4\right)}+X_{43}^{\left(5\right)}& X_{10,8}^{\left(4\right)}& X_{10,10}^{\left(4\right)}+X_{44}^{\left(5\right)}\\ X_{11,1}^{\left(4\right)}& X_{11,2}^{\left(4\right)}& X_{11,4}^{\left(4\right)}& X_{11,5}^{\left(4\right)}& X_{11,7}^{\left(4\right)}& X_{11,8}^{\left(4\right)}& X_{11,10}^{\left(4\right)}\\ X_{13,1}^{\left(4\right)}+X_{51}^{\left(5\right)}& X_{13,2}^{\left(4\right)}& X_{13,4}^{\left(4\right)}+X_{52}^{\left(5\right)}& X_{13,5}^{\left(4\right)}& X_{13,7}^{\left(4\right)}+X_{53}^{\left(5\right)}& X_{13,8}^{\left(4\right)}& X_{13,10}^{\left(4\right)}+X_{54}^{\left(5\right)}\\ X_{14,1}^{\left(4\right)}& X_{14,2}^{\left(4\right)}& X_{14,4}^{\left(4\right)}& X_{14,5}^{\left(4\right)}& X_{14,7}^{\left(4\right)}& X_{14,8}^{\left(4\right)}& X_{14,10}^{\left(4\right)}\\ X_{16,1}^{\left(4\right)}+X_{61}^{\left(5\right)}& X_{16,2}^{\left(4\right)}& X_{16,4}^{\left(4\right)}+X_{62}^{\left(5\right)}& X_{16,5}^{\left(4\right)}& X_{16,7}^{\left(4\right)}+X_{63}^{\left(5\right)}& X_{16,8}^{\left(4\right)}& X_{16,10}^{\left(4\right)}+X_{64}^{\left(5\right)}\\ X_{17,1}^{\left(4\right)}& X_{17,2}^{\left(4\right)}& X_{17,4}^{\left(4\right)}& X_{17,5}^{\left(4\right)}& X_{17,7}^{\left(4\right)}& X_{17,8}^{\left(4\right)}& X_{17,10}^{\left(4\right)}\\ X_{19,1}^{\left(4\right)}+X_{71}^{\left(5\right)}& X_{19,2}^{\left(4\right)}& X_{19,4}^{\left(4\right)}+X_{72}^{\left(5\right)}& X_{19,5}^{\left(4\right)}& X_{19,7}^{\left(4\right)}+X_{73}^{\left(5\right)}& X_{19,8}^{\left(4\right)}& X_{19,10}^{\left(4\right)}+X_{74}^{\left(5\right)}\\ X_{20,1}^{\left(4\right)}& X_{20,2}^{\left(4\right)}& X_{20,4}^{\left(4\right)}& X_{20,5}^{\left(4\right)}& X_{20,7}^{\left(4\right)}& X_{20,8}^{\left(4\right)}& X_{20,10}^{\left(4\right)}\\ X_{81}^{\left(5\right)}& 0& X_{82}^{\left(5\right)}& 0& X_{83}^{\left(5\right)}& 0& X_{84}^{\left(5\right)}\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\Phi }_2^{\left(4\right)}=\left(\begin{array}{ccccccccc}{X}_{1,11}^{\left(4\right)}& X_{1,13}^{\left(4\right)}+X_{15}^{\left(5\right)}& X_{1,14}^{\left(4\right)}& X_{1,16}^{\left(4\right)}+X_{16}^{\left(5\right)}& X_{1,17}^{\left(4\right)}& X_{1,19}^{\left(4\right)}+X_{17}^{\left(5\right)}& X_{1,20}^{\left(4\right)}& X_{18}^{\left(5\right)}& 0\\ X_{2,11}^{\left(4\right)}& X_{2,13}^{\left(4\right)}& X_{2,14}^{\left(4\right)}& X_{2,16}^{\left(4\right)}& X_{2,17}^{\left(4\right)}& X_{2,19}^{\left(4\right)}& X_{2,20}^{\left(4\right)}& 0& 0\\ X_{4,11}^{\left(4\right)}& X_{4,13}^{\left(4\right)}+X_{25}^{\left(5\right)}& X_{4,14}^{\left(4\right)}& X_{4,16}^{\left(4\right)}+X_{26}^{\left(5\right)}& X_{4,17}^{\left(4\right)}& X_{4,19}^{\left(4\right)}+X_{27}^{\left(5\right)}& X_{4,20}^{\left(4\right)}& X_{28}^{\left(5\right)}& 0\\ X_{5,11}^{\left(4\right)}& X_{5,13}^{\left(4\right)}& X_{5,14}^{\left(4\right)}& X_{5,16}^{\left(4\right)}& X_{5,17}^{\left(4\right)}& X_{5,19}^{\left(4\right)}& X_{5,20}^{\left(4\right)}& 0& 0\\ X_{7,11}^{\left(4\right)}& X_{7,13}^{\left(4\right)}+X_{35}^{\left(5\right)}& X_{7,14}^{\left(4\right)}& X_{7,16}^{\left(4\right)}+X_{36}^{\left(5\right)}& X_{7,17}^{\left(4\right)}& X_{7,19}^{\left(4\right)}+X_{37}^{\left(5\right)}& X_{7,20}^{\left(4\right)}& X_{38}^{\left(5\right)}& 0\\ X_{8,11}^{\left(4\right)}& X_{8,13}^{\left(4\right)}& X_{8,14}^{\left(4\right)}& X_{8,16}^{\left(4\right)}& X_{8,17}^{\left(4\right)}& X_{8,19}^{\left(4\right)}& X_{8,20}^{\left(4\right)}& 0& 0\\ X_{10,11}^{\left(4\right)}& X_{10,13}^{\left(4\right)}+X_{45}^{\left(5\right)}& X_{10,14}^{\left(4\right)}& X_{10,16}^{\left(4\right)}+X_{46}^{\left(5\right)}& X_{10,17}^{\left(4\right)}& X_{10,19}^{\left(4\right)}+X_{47}^{\left(5\right)}& X_{10,20}^{\left(4\right)}& X_{48}^{\left(5\right)}& 0\\ X_{11,11}^{\left(4\right)}& X_{11,13}^{\left(4\right)}& X_{11,14}^{\left(4\right)}& X_{11,16}^{\left(4\right)}& X_{11,17}^{\left(4\right)}& X_{11,19}^{\left(4\right)}& X_{11,20}^{\left(4\right)}& 0& 0\\ X_{13,11}^{\left(4\right)}& X_{13,13}^{\left(4\right)}+X_{55}^{\left(5\right)}& X_{13,14}^{\left(4\right)}& X_{13,16}^{\left(4\right)}+X_{56}^{\left(5\right)}& X_{13,17}^{\left(4\right)}& X_{13,19}^{\left(4\right)}+X_{57}^{\left(5\right)}& X_{13,20}^{\left(4\right)}& X_{58}^{\left(5\right)}& 0\\ X_{14,11}^{\left(4\right)}& X_{14,13}^{\left(4\right)}& X_{14,14}^{\left(4\right)}& X_{14,16}^{\left(4\right)}& X_{14,17}^{\left(4\right)}& X_{14,19}^{\left(4\right)}& X_{14,20}^{\left(4\right)}& 0& 0\\ X_{16,11}^{\left(4\right)}& X_{16,13}^{\left(4\right)}+X_{65}^{\left(5\right)}& X_{16,14}^{\left(4\right)}& X_{16,16}^{\left(4\right)}+X_{66}^{\left(5\right)}& X_{16,17}^{\left(4\right)}& X_{16,19}^{\left(4\right)}+X_{67}^{\left(5\right)}& X_{16,20}^{\left(4\right)}& X_{68}^{\left(5\right)}& 0\\ X_{17,11}^{\left(4\right)}& X_{17,13}^{\left(4\right)}& X_{17,14}^{\left(4\right)}& X_{17,16}^{\left(4\right)}& X_{17,17}^{\left(4\right)}& X_{17,19}^{\left(4\right)}& X_{17,20}^{\left(4\right)}& 0& 0\\ X_{19,11}^{\left(4\right)}& X_{19,13}^{\left(4\right)}+X_{75}^{\left(5\right)}& X_{19,14}^{\left(4\right)}& X_{19,16}^{\left(4\right)}+X_{76}^{\left(5\right)}& X_{19,17}^{\left(4\right)}& X_{19,19}^{\left(4\right)}+X_{77}^{\left(5\right)}& X_{19,20}^{\left(4\right)}& X_{78}^{\left(5\right)}& 0\\ X_{20,11}^{\left(4\right)}& X_{20,13}^{\left(4\right)}& X_{20,14}^{\left(4\right)}& X_{20,16}^{\left(4\right)}& X_{20,17}^{\left(4\right)}& X_{20,19}^{\left(4\right)}& X_{20,20}^{\left(4\right)}& 0& 0\\ 0& X_{85}^{\left(5\right)}& 0& X_{86}^{\left(5\right)}& 0& X_{87}^{\left(5\right)}& 0& X_{88}^{\left(5\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right).\end{array}$$

 Theorem 1. 

In terms of system (1), the following conditions are equivalent to each other:

1.

System (1) is consistent.

2.

The ranks of $A_i$, $B_i$, $C_i$, $D_i$, ${\Omega }_i$, $i=1,\cdots ,4$ satisfy the following 40 rank equalities:

$$\begin{array}{c}\hfill r_{A_i{\Omega }_i{B}_i}=r_{A_i{B}_i},\end{array}$$

(12)

$$\begin{array}{c}\hfill r_{C_i\left|{\Omega }_i\right|D_i}=r_{C_i|D_i},\end{array}$$

(13)

$$\begin{array}{c}\hfill r_{A_i{\Omega }_i|0D_i}=r_{A_i}+r_{D_i},\end{array}$$

(14)

$$\begin{array}{c}\hfill r_{C_{i}0|{\Omega }_i{B}_i}=r_{C_i}+r_{B_i},\end{array}$$

(15)

$$\begin{array}{c}\hfill r_{A_1{\Omega }_1{B}_{1}00\left|0D_{1}0C_{2}0\right|00A_2-{\Omega }_2{B}_2}=r_{A_1{B}_{1}0|0A_2{B}_2}+r_{D_1{C}_2},\end{array}$$

(16)

$$\begin{array}{c}\hfill r_{C_{1}00|{\Omega }_1{B}_{1}0|D_{1}0C_2|0A_2-{\Omega }_2|00D_2}=r_{B_1|A_2}+r_{C_{1}0\left|D_1{C}_2\right|0D_2},\end{array}$$

(17)

$$\begin{array}{c}\hfill r_{A_1{\Omega }_1{B}_{1}0|0D_{1}0C_2|00A_2-{\Omega }_2|000D_2}=r_{A_1{B}_1|0A_2}+r_{D_1{C}_2|0D_2},\end{array}$$

(18)

$$\begin{array}{c}\hfill r_{C_{1}000|{\Omega }_1{B}_{1}00\left|D_{1}0C_{2}0\right|0A_2-{\Omega }_2{B}_2}=r_{B_{1}0|A_2{B}_2}+r_{C_{1}0|D_1{C}_2},\end{array}$$

(19)

$$\begin{array}{c}\hfill r_{A_2{\Omega }_2{B}_{2}00\left|0D_{2}0C_{3}0\right|00A_3-{\Omega }_3{B}_3}=r_{A_2{B}_{2}0|0A_3{B}_3}+r_{D_2{C}_3},\end{array}$$

(20)

$$\begin{array}{c}\hfill r_{C_{2}00|{\Omega }_2{B}_{2}0|D_{2}0C_3|0A_3-{\Omega }_3|00D_3}=r_{B_2|A_3}+r_{C_{2}0\left|D_2{C}_3\right|0D_3},\end{array}$$

(21)

$$\begin{array}{c}\hfill r_{A_2{\Omega }_2{B}_{2}0|0D_{2}0C_3|00A_3-{\Omega }_3|000D_3}=r_{A_2{B}_2|0A_3}+r_{D_2{C}_3|0D_3},\end{array}$$

(22)

$$\begin{array}{c}\hfill r_{C_{2}000|{\Omega }_2{B}_{2}00\left|D_{2}0C_{3}0\right|0A_3-{\Omega }_3{B}_3}=r_{B_{2}0|A_3{B}_3}+r_{C_{2}0|D_2{C}_3},\end{array}$$

(23)

$$\begin{array}{c}\hfill r_{A_3{\Omega }_3{B}_{3}00\left|0D_{3}0C_{4}0\right|00A_4-{\Omega }_4{B}_4}=r_{A_3{B}_{3}0|0A_4{B}_4}+r_{D_3{C}_4},\end{array}$$

(24)

$$\begin{array}{c}\hfill r_{C_{3}00|{\Omega }_3{B}_{3}0|D_{3}0C_4|0A_4-{\Omega }_4|00D_4}=r_{B_3|A_4}+r_{C_{3}0\left|D_3{C}_4\right|0D_4},\end{array}$$

(25)

$$\begin{array}{c}\hfill r_{A_3{\Omega }_3{B}_{3}0|0D_{3}0C_4|00A_4-{\Omega }_4|000D_4}=r_{A_3{B}_3|0A_4}+r_{D_3{C}_4|0D_4},\end{array}$$

(26)

$$\begin{array}{c}\hfill r_{C_{3}000|{\Omega }_3{B}_{3}00\left|D_{3}0C_{4}0\right|0A_4-{\Omega }_4{B}_4}=r_{B_{3}0|A_4{B}_4}+r_{C_{3}0|D_3{C}_4},\end{array}$$

(27)

$$\begin{array}{c}\hfill r_{A_1{\Omega }_1{B}_{1}0000|0D_{1}0C_{2}000|00A_2-{\Omega }_2{B}_{2}00\left|000D_{2}0C_{3}0\right|0000A_3{\Omega }_3{B}_3}=r_{A_1{B}_{1}00\left|0A_2{B}_{2}0\right|00A_3{B}_3}+r_{D_1{C}_{2}0|0D_2{C}_3},\end{array}$$

(28)

$$\begin{array}{c}\hfill r_{C_{1}0000|{\Omega }_1{B}_{1}000|D_{1}0C_{2}00|0A_2-{\Omega }_2{B}_{2}0|00D_{2}0C_3\left|000A_3{\Omega }_3\right|0000D_3}=r_{B_{1}0\left|A_2{B}_2\right|0A_3}+r_{C_{1}00|D_1{C}_{2}0\left|0D_2{C}_3\right|00D_3},\end{array}$$

(29)

$$\begin{array}{c}\hfill r_{A_1{\Omega }_1{B}_{1}000|0D_{1}0C_{2}00|00A_2-{\Omega }_2{B}_{2}0|000D_{2}0C_3\left|0000A_3{\Omega }_3\right|00000D_3}=r_{A_1{B}_{1}0\left|0A_2{B}_2\right|00A_3}+r_{D_1{C}_{2}0\left|0D_2{C}_3\right|00D_3},\end{array}$$

(30)

$$\begin{array}{c}\hfill r_{C_{1}00000|{\Omega }_1{B}_{1}0000|D_{1}0C_{2}000|0A_2-{\Omega }_2{B}_{2}00\left|00D_{2}0C_{3}0\right|000A_3{\Omega }_3{B}_3}=r_{B_{1}00\left|A_2{B}_{2}0\right|0A_3{B}_3}+r_{C_{1}00\left|D_1{C}_{2}0\right|0D_2{C}_3},\end{array}$$

(31)

$$\begin{array}{c}\hfill r_{A_2{\Omega }_2{B}_{2}0000|0D_{2}0C_{3}000|00A_3-{\Omega }_3{B}_{3}00\left|000D_{3}0C_{4}0\right|0000A_4{\Omega }_4{B}_4}=r_{A_2{B}_{2}00\left|0A_3{B}_{3}0\right|00A_4{B}_4}+r_{D_2{C}_{3}0|0D_3{C}_4},\end{array}$$

(32)

$$\begin{array}{c}\hfill r_{C_{2}0000|{\Omega }_2{B}_{2}000|D_{2}0C_{3}00|0A_3-{\Omega }_3{B}_{3}0|00D_{3}0C_4\left|000A_4{\Omega }_4\right|0000D_4}=r_{B_{2}0\left|A_3{B}_3\right|0A_4}+r_{C_{2}00|D_2{C}_{3}0\left|0D_3{C}_4\right|00D_4},\end{array}$$

(33)

$$\begin{array}{c}\hfill r_{A_2{\Omega }_2{B}_{2}000|0D_{2}0C_{3}00|00A_3-{\Omega }_3{B}_{3}0|000D_{3}0C_4\left|0000A_4{\Omega }_4\right|00000D_4}=r_{A_2{B}_{2}0\left|0A_3{B}_3\right|00A_4}+r_{D_2{C}_{3}0\left|0D_3{C}_4\right|00D_4},\end{array}$$

(34)

$$\begin{array}{c}\hfill r_{C_{2}00000|{\Omega }_2{B}_{2}0000|D_{2}0C_{3}000|0A_3-{\Omega }_3{B}_{3}00\left|00D_{3}0C_{4}0\right|000A_4{\Omega }_4{B}_4}=r_{B_{2}00\left|A_3{B}_{3}0\right|0A_4{B}_4}+r_{C_{2}00\left|D_2{C}_{3}0\right|0D_3{C}_4},\end{array}$$

(35)

$$\begin{array}{cc}\hfill & r_{A_1{\Omega }_1{B}_{1}000000|0D_{1}0C_{2}00000|00A_2-{\Omega }_2{B}_{2}0000|000D_{2}0C_{3}000|0000A_3{\Omega }_3{B}_{3}00\left|00000D_{3}0C_{4}0\right|000000A_4-{\Omega }_4{B}_4}\hfill \\ \hfill & =r_{A_1{B}_{1}000|0A_2{B}_{2}00\left|00A_3{B}_{3}0\right|000A_4{B}_4}+r_{D_1{C}_{2}00\left|0D_2{C}_{3}0\right|00D_3{C}_4},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & r_{C_{1}000000|{\Omega }_1{B}_{1}00000|D_{1}0C_{2}0000|0A_2-{\Omega }_2{B}_{2}000|00D_{2}0C_{3}00|000A_3{\Omega }_3{B}_{3}0|0000D_{3}0C_4|00000A_4-{\Omega }_4|000000D_4}\hfill \\ \hfill & =r_{B_{1}00|A_2{B}_{2}0\left|0A_3{B}_3\right|00A_4}+r_{C_{1}000|D_1{C}_{2}00|0D_2{C}_{3}0\left|00D_3{C}_4\right|000D_4},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & r_{A_1{\Omega }_1{B}_{1}00000|0D_{1}0C_{2}0000|00A_2-{\Omega }_2{B}_{2}000|000D_{2}0C_{3}00|0000A_3{\Omega }_3{B}_{3}0|00000D_{3}0C_4|000000A_4-{\Omega }_4|0000000D_4}\hfill \\ \hfill & =r_{A_1{B}_{1}00|0A_2{B}_{2}0\left|00A_3{B}_3\right|000A_4}+r_{D_1{C}_{2}00|0D_2{C}_{3}0\left|00D_3{C}_4\right|000D_4},\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & r_{C_{1}0000000|{\Omega }_1{B}_{1}000000|D_{1}0C_{2}00000|0A_2-{\Omega }_2{B}_{2}0000|00D_{2}0C_{3}000|000A_3{\Omega }_3{B}_{3}00\left|0000D_{3}0C_{4}0\right|00000A_4-{\Omega }_4{B}_4}\hfill \\ \hfill & =r_{B_{1}000|A_2{B}_{2}00\left|0A_3{B}_{3}0\right|00A_4{B}_4}+r_{C_{1}000|D_1{C}_{2}00\left|0D_2{C}_{3}0\right|00D_3{C}_4}.\hfill \end{array}$$

(39)

3.

The block matrices satisfy

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(\begin{array}{cccc}{\omega }_{16,1}^{\left(1\right)}& {\omega }_{16,2}^{\left(1\right)}& \cdots & {\omega }_{16,16}^{\left(1\right)}\end{array}\right)=0,\\ \left(\begin{array}{cccc}{\omega }_{24,1}^{\left(2\right)}& {\omega }_{24,2}^{\left(2\right)}& \cdots & {\omega }_{24,24}^{\left(2\right)}\end{array}\right)=0,\\ \left(\begin{array}{cccc}{\omega }_{24,1}^{\left(3\right)}& {\omega }_{24,2}^{\left(3\right)}& \cdots & {\omega }_{24,24}^{\left(3\right)}\end{array}\right)=0,\\ \left(\begin{array}{cccc}{\omega }_{16,1}^{\left(4\right)}& {\omega }_{16,2}^{\left(4\right)}& \cdots & {\omega }_{16,16}^{\left(4\right)}\end{array}\right)=0,\end{array}\right.\left(\begin{array}{c}{\omega }_{1,16}^{\left(1\right)}\\ {\omega }_{2,16}^{\left(1\right)}\\ ⋮\\ {\omega }_{16,16}^{\left(1\right)}\end{array}\right)=0,\left(\begin{array}{c}{\omega }_{1,24}^{\left(2\right)}\\ {\omega }_{2,24}^{\left(2\right)}\\ ⋮\\ {\omega }_{24,24}^{\left(2\right)}\end{array}\right)=0,\left(\begin{array}{c}{\omega }_{1,24}^{\left(3\right)}\\ {\omega }_{2,24}^{\left(3\right)}\\ ⋮\\ {\omega }_{24,24}^{\left(3\right)}\end{array}\right)=0,\left(\begin{array}{c}{\omega }_{1,16}^{\left(4\right)}\\ {\omega }_{2,16}^{\left(4\right)}\\ ⋮\\ {\omega }_{16,16}^{\left(4\right)}\end{array}\right)=0,\end{array}$$

(40)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{98}^{\left(1\right)}=0,{\omega }_{10,8}^{\left(1\right)}=0,{\omega }_{11,8}^{\left(1\right)}=0,{\omega }_{12,8}^{\left(1\right)}=0,{\omega }_{13,8}^{\left(1\right)}=0,{\omega }_{14,8}^{\left(1\right)}=0,{\omega }_{15,8}^{\left(1\right)}=0,\hfill \\ {\omega }_{19,6}^{\left(2\right)}=0,{\omega }_{20,6}^{\left(2\right)}=0,{\omega }_{21,6}^{\left(2\right)}=0,{\omega }_{22,6}^{\left(2\right)}=0,{\omega }_{23,6}^{\left(2\right)}=0,{\omega }_{19,12}^{\left(2\right)}=0,{\omega }_{20,12}^{\left(2\right)}=0,{\omega }_{21,12}^{\left(2\right)}=0,\hfill \\ {\omega }_{22,12}^{\left(2\right)}=0,{\omega }_{23,12}^{\left(2\right)}=0,{\omega }_{19,18}^{\left(2\right)}=0,{\omega }_{20,18}^{\left(2\right)}=0,{\omega }_{21,18}^{\left(2\right)}=0,{\omega }_{22,18}^{\left(2\right)}=0,{\omega }_{23,18}^{\left(2\right)}=0,\hfill \\ {\omega }_{21,4}^{\left(3\right)}=0,{\omega }_{22,4}^{\left(3\right)}=0,{\omega }_{23,4}^{\left(3\right)}=0,{\omega }_{21,8}^{\left(3\right)}=0,{\omega }_{22,8}^{\left(3\right)}=0,{\omega }_{23,8}^{\left(3\right)}=0,{\omega }_{21,12}^{\left(3\right)}=0,{\omega }_{22,12}^{\left(3\right)}=0,\hfill \\ {\omega }_{23,12}^{\left(3\right)}=0,{\omega }_{21,16}^{\left(3\right)}=0,{\omega }_{22,16}^{\left(3\right)}=0,{\omega }_{23,16}^{\left(3\right)}=0,{\omega }_{21,20}^{\left(3\right)}=0,{\omega }_{22,20}^{\left(3\right)}=0,{\omega }_{23,20}^{\left(3\right)}=0,\hfill \\ {\omega }_{15,2}^{\left(4\right)}=0,{\omega }_{15,4}^{\left(4\right)}=0,{\omega }_{15,6}^{\left(4\right)},{\omega }_{15,8}^{\left(4\right)}=0,{\omega }_{15,10}^{\left(4\right)}=0,{\omega }_{15,12}^{\left(4\right)}=0,{\omega }_{15,14}^{\left(4\right)}=0,\hfill \end{array}\right.\end{array}$$

(41)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{89}^{\left(1\right)}=0,{\omega }_{8,10}^{\left(1\right)}=0,{\omega }_{8,11}^{\left(1\right)}=0,{\omega }_{8,12}^{\left(1\right)}=0,{\omega }_{8,13}^{\left(1\right)}=0,{\omega }_{8,14}^{\left(1\right)}=0,{\omega }_{8,15}^{\left(1\right)}=0,\hfill \\ {\omega }_{6,19}^{\left(2\right)}=0,{\omega }_{6,20}^{\left(2\right)}=0,{\omega }_{6,21}^{\left(2\right)}=0,{\omega }_{6,22}^{\left(2\right)}=0,{\omega }_{6,23}^{\left(2\right)}=0,{\omega }_{12,19}^{\left(2\right)}=0,{\omega }_{12,20}^{\left(2\right)}=0,{\omega }_{12,21}^{\left(2\right)}=0,\hfill \\ {\omega }_{12,22}^{\left(2\right)}=0,{\omega }_{12,23}^{\left(2\right)}=0,{\omega }_{18,19}^{\left(2\right)}=0,{\omega }_{18,20}^{\left(2\right)}=0,{\omega }_{18,21}^{\left(2\right)},{\omega }_{18,22}^{\left(2\right)}=0,{\omega }_{18,23}^{\left(2\right)}=0,\hfill \\ {\omega }_{4,21}^{\left(3\right)}=0,{\omega }_{4,22}^{\left(3\right)}=0,{\omega }_{4,23}^{\left(3\right)}=0,{\omega }_{8,21}^{\left(3\right)}=0,{\omega }_{8,22}^{\left(3\right)}=0,{\omega }_{8,23}^{\left(3\right)}=0,{\omega }_{12,21}^{\left(3\right)}=0,{\omega }_{12,22}^{\left(3\right)}=0,\hfill \\ {\omega }_{12,23}^{\left(3\right)}=0,{\omega }_{16,21}^{\left(3\right)}=0,{\omega }_{16,22}^{\left(3\right)}=0,{\omega }_{16,23}^{\left(3\right)}=0,{\omega }_{20,21}^{\left(3\right)}=0,{\omega }_{20,22}^{\left(3\right)}=0,{\omega }_{20,23}^{\left(3\right)}=0,\hfill \\ {\omega }_{2,15}^{\left(4\right)}=0,{\omega }_{4,15}^{\left(4\right)}=0,{\omega }_{6,15}^{\left(4\right)}=0,{\omega }_{8,15}^{\left(4\right)}=0,{\omega }_{10,15}^{\left(4\right)}=0,{\omega }_{12,15}^{\left(4\right)}=0,{\omega }_{14,15}^{\left(4\right)}=0,\hfill \end{array}\right.\end{array}$$

(42)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{14,1}^{\left(1\right)}={\omega }_{12,1}^{\left(2\right)},{\omega }_{14,2}^{\left(1\right)}={\omega }_{12,2}^{\left(2\right)},{\omega }_{14,3}^{\left(1\right)}={\omega }_{12,3}^{\left(2\right)},{\omega }_{14,4}^{\left(1\right)}={\omega }_{12,4}^{\left(2\right)},{\omega }_{14,5}^{\left(1\right)}={\omega }_{12,5}^{\left(2\right)},{\omega }_{14,6}^{\left(1\right)}={\omega }_{12,6}^{\left(2\right)},\hfill \\ {\omega }_{14,9}^{\left(1\right)}={\omega }_{12,7}^{\left(2\right)},{\omega }_{14,10}^{\left(1\right)}={\omega }_{12,8}^{\left(2\right)},{\omega }_{14,11}^{\left(1\right)}={\omega }_{12,9}^{\left(2\right)},{\omega }_{14,12}^{\left(1\right)}={\omega }_{12,10}^{\left(2\right)},{\omega }_{14,13}^{\left(1\right)}={\omega }_{12,11}^{\left(2\right)},{\omega }_{14,14}^{\left(1\right)}={\omega }_{12,12}^{\left(2\right)},\hfill \end{array}\right.\end{array}$$

(43)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{1,14}^{\left(1\right)}={\omega }_{1,12}^{\left(2\right)},{\omega }_{2,14}^{\left(1\right)}={\omega }_{2,12}^{\left(2\right)},{\omega }_{3,14}^{\left(1\right)}={\omega }_{3,12}^{\left(2\right)},{\omega }_{4,14}^{\left(1\right)}={\omega }_{4,12}^{\left(2\right)},{\omega }_{5,14}^{\left(1\right)}={\omega }_{5,12}^{\left(2\right)},{\omega }_{6,14}^{\left(1\right)}={\omega }_{6,12}^{\left(2\right)},\hfill \\ {\omega }_{9,14}^{\left(1\right)}={\omega }_{7,12}^{\left(2\right)},{\omega }_{10,14}^{\left(1\right)}={\omega }_{8,12}^{\left(2\right)},{\omega }_{11,14}^{\left(1\right)}={\omega }_{9,12}^{\left(2\right)},{\omega }_{12,14}^{\left(1\right)}={\omega }_{10,12}^{\left(2\right)},{\omega }_{13,14}^{\left(1\right)}={\omega }_{11,12}^{\left(2\right)},{\omega }_{14,14}^{\left(1\right)}={\omega }_{12,12}^{\left(2\right)},\hfill \end{array}\right.\end{array}$$

(44)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{9,6}^{\left(1\right)}={\omega }_{76}^{\left(2\right)},{\omega }_{10,6}^{\left(1\right)}={\omega }_{86}^{\left(2\right)},{\omega }_{11,6}^{\left(1\right)}={\omega }_{96}^{\left(2\right)},{\omega }_{12,6}^{\left(1\right)}={\omega }_{10,6}^{\left(2\right)},{\omega }_{13,6}^{\left(1\right)}={\omega }_{11,6}^{\left(2\right)},{\omega }_{14,6}^{\left(1\right)}={\omega }_{12,6}^{\left(2\right)},\hfill \\ {\omega }_{9,14}^{\left(1\right)}={\omega }_{7,12}^{\left(2\right)},{\omega }_{10,14}^{\left(1\right)}={\omega }_{8,12}^{\left(2\right)},{\omega }_{11,14}^{\left(1\right)}={\omega }_{9,12}^{\left(2\right)},{\omega }_{12,14}^{\left(1\right)}={\omega }_{10,12}^{\left(2\right)},{\omega }_{13,14}^{\left(1\right)}={\omega }_{11,12}^{\left(2\right)},{\omega }_{14,14}^{\left(1\right)}={\omega }_{12,12}^{\left(2\right)},\hfill \end{array}\right.\end{array}$$

(45)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{6,9}^{\left(1\right)}={\omega }_{67}^{\left(2\right)},{\omega }_{6,10}^{\left(1\right)}={\omega }_{68}^{\left(2\right)},{\omega }_{6,11}^{\left(1\right)}={\omega }_{69}^{\left(2\right)},{\omega }_{6,12}^{\left(1\right)}={\omega }_{6,10}^{\left(2\right)},{\omega }_{6,13}^{\left(1\right)}={\omega }_{6,11}^{\left(2\right)},{\omega }_{6,14}^{\left(1\right)}={\omega }_{6,12}^{\left(2\right)},\hfill \\ {\omega }_{14,9}^{\left(1\right)}={\omega }_{12,7}^{\left(2\right)},{\omega }_{14,10}^{\left(1\right)}={\omega }_{12,8}^{\left(2\right)},{\omega }_{14,11}^{\left(1\right)}={\omega }_{12,9}^{\left(2\right)},{\omega }_{14,12}^{\left(1\right)}={\omega }_{12,10}^{\left(2\right)},{\omega }_{14,13}^{\left(1\right)}={\omega }_{12,11}^{\left(2\right)},{\omega }_{14,14}^{\left(1\right)}={\omega }_{12,12}^{\left(2\right)},\hfill \end{array}\right.\end{array}$$

(46)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{22,1}^{\left(2\right)}={\omega }_{16,1}^{\left(3\right)},{\omega }_{22,2}^{\left(2\right)}={\omega }_{16,2}^{\left(3\right)},{\omega }_{22,3}^{\left(2\right)}={\omega }_{16,3}^{\left(3\right)},{\omega }_{22,4}^{\left(2\right)}={\omega }_{16,4}^{\left(3\right)},{\omega }_{22,7}^{\left(2\right)}={\omega }_{16,5}^{\left(3\right)},{\omega }_{22,8}^{\left(2\right)}={\omega }_{16,6}^{\left(3\right)},\hfill \\ {\omega }_{22,9}^{\left(2\right)}={\omega }_{16,7}^{\left(3\right)},{\omega }_{22,10}^{\left(2\right)}={\omega }_{16,8}^{\left(3\right)},{\omega }_{22,13}^{\left(2\right)}={\omega }_{16,9}^{\left(3\right)},{\omega }_{22,14}^{\left(2\right)}={\omega }_{16,10}^{\left(3\right)},{\omega }_{22,15}^{\left(2\right)}={\omega }_{16,11}^{\left(3\right)},{\omega }_{22,16}^{\left(2\right)}={\omega }_{16,12}^{\left(3\right)},\hfill \\ {\omega }_{22,19}^{\left(2\right)}={\omega }_{16,13}^{\left(3\right)},{\omega }_{22,20}^{\left(2\right)}={\omega }_{16,14}^{\left(3\right)},{\omega }_{22,21}^{\left(2\right)}={\omega }_{16,15}^{\left(3\right)},{\omega }_{22,22}^{\left(2\right)}={\omega }_{16,16}^{\left(3\right)},\hfill \end{array}\right.\end{array}$$

(47)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{1,22}^{\left(2\right)}={\omega }_{1,16}^{\left(3\right)},{\omega }_{2,22}^{\left(2\right)}={\omega }_{2,16}^{\left(3\right)},{\omega }_{3,22}^{\left(2\right)}={\omega }_{3,16}^{\left(3\right)},{\omega }_{4,22}^{\left(2\right)}={\omega }_{4,16}^{\left(3\right)},{\omega }_{7,22}^{\left(2\right)}={\omega }_{5,16}^{\left(3\right)},{\omega }_{8,22}^{\left(2\right)}={\omega }_{6,16}^{\left(3\right)},\hfill \\ {\omega }_{9,22}^{\left(2\right)}={\omega }_{7,16}^{\left(3\right)},{\omega }_{10,22}^{\left(2\right)}={\omega }_{8,16}^{\left(3\right)},{\omega }_{13,22}^{\left(2\right)}={\omega }_{9,16}^{\left(3\right)},{\omega }_{14,22}^{\left(2\right)}={\omega }_{10,16}^{\left(3\right)},{\omega }_{15,22}^{\left(2\right)}={\omega }_{11,16}^{\left(3\right)},{\omega }_{16,22}^{\left(2\right)}={\omega }_{12,16}^{\left(3\right)},\hfill \\ {\omega }_{19,22}^{\left(2\right)}={\omega }_{13,16}^{\left(3\right)},{\omega }_{20,22}^{\left(2\right)}={\omega }_{14,16}^{\left(3\right)},{\omega }_{21,22}^{\left(2\right)}={\omega }_{15,16}^{\left(3\right)},{\omega }_{22,22}^{\left(2\right)}={\omega }_{16,16}^{\left(3\right)},\hfill \end{array}\right.\end{array}$$

(48)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{19,4}^{\left(2\right)}={\omega }_{13,4}^{\left(3\right)},{\omega }_{20,4}^{\left(2\right)}={\omega }_{14,4}^{\left(3\right)},{\omega }_{21,4}^{\left(2\right)}={\omega }_{15,4}^{\left(3\right)},{\omega }_{22,4}^{\left(2\right)}={\omega }_{16,4}^{\left(3\right)},{\omega }_{19,10}^{\left(2\right)}={\omega }_{13,8}^{\left(3\right)},{\omega }_{20,10}^{\left(2\right)}={\omega }_{14,8}^{\left(3\right)},\hfill \\ {\omega }_{21,10}^{\left(2\right)}={\omega }_{15,8}^{\left(3\right)},{\omega }_{22,10}^{\left(2\right)}={\omega }_{16,8}^{\left(3\right)},{\omega }_{19,16}^{\left(2\right)}={\omega }_{13,12}^{\left(3\right)},{\omega }_{20,16}^{\left(2\right)}={\omega }_{14,12}^{\left(3\right)},{\omega }_{21,16}^{\left(2\right)}={\omega }_{15,12}^{\left(3\right)},{\omega }_{22,16}^{\left(2\right)}={\omega }_{16,12}^{\left(3\right)},\hfill \\ {\omega }_{19,22}^{\left(2\right)}={\omega }_{13,16}^{\left(3\right)},{\omega }_{20,22}^{\left(2\right)}={\omega }_{14,16}^{\left(3\right)},{\omega }_{21,22}^{\left(2\right)}={\omega }_{15,16}^{\left(3\right)},{\omega }_{22,22}^{\left(2\right)}={\omega }_{16,16}^{\left(3\right)},\hfill \end{array}\right.\end{array}$$

(49)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{4,19}^{\left(2\right)}={\omega }_{4,13}^{\left(3\right)},{\omega }_{4,20}^{\left(2\right)}={\omega }_{4,14}^{\left(3\right)},{\omega }_{4,21}^{\left(2\right)}={\omega }_{4,15}^{\left(3\right)},{\omega }_{4,22}^{\left(2\right)}={\omega }_{4,16}^{\left(3\right)},{\omega }_{10,19}^{\left(2\right)}={\omega }_{8,13}^{\left(3\right)},{\omega }_{10,20}^{\left(2\right)}={\omega }_{8,14}^{\left(3\right)},\hfill \\ {\omega }_{10,21}^{\left(2\right)}={\omega }_{8,15}^{\left(3\right)},{\omega }_{10,22}^{\left(2\right)}={\omega }_{8,16}^{\left(3\right)},{\omega }_{16,19}^{\left(2\right)}={\omega }_{12,13}^{\left(3\right)},{\omega }_{16,20}^{\left(2\right)}={\omega }_{12,14}^{\left(3\right)},{\omega }_{16,21}^{\left(2\right)}={\omega }_{12,15}^{\left(3\right)},{\omega }_{16,22}^{\left(2\right)}={\omega }_{12,16}^{\left(3\right)},\hfill \\ {\omega }_{22,19}^{\left(2\right)}={\omega }_{16,13}^{\left(3\right)},{\omega }_{22,20}^{\left(2\right)}={\omega }_{16,14}^{\left(3\right)},{\omega }_{22,21}^{\left(2\right)}={\omega }_{16,15}^{\left(3\right)},{\omega }_{22,22}^{\left(2\right)}={\omega }_{16,16}^{\left(3\right)},\hfill \end{array}\right.\end{array}$$

(50)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{22,1}^{\left(3\right)}={\omega }_{12,1}^{\left(4\right)},{\omega }_{22,2}^{\left(3\right)}={\omega }_{12,2}^{\left(4\right)},{\omega }_{22,5}^{\left(3\right)}={\omega }_{12,3}^{\left(4\right)},{\omega }_{22,6}^{\left(3\right)}={\omega }_{12,4}^{\left(4\right)},{\omega }_{22,9}^{\left(3\right)}={\omega }_{12,5}^{\left(4\right)},{\omega }_{22,10}^{\left(3\right)}={\omega }_{12,6}^{\left(4\right)},\hfill \\ {\omega }_{22,13}^{\left(3\right)}={\omega }_{12,7}^{\left(4\right)},{\omega }_{22,14}^{\left(3\right)}={\omega }_{12,8}^{\left(4\right)},{\omega }_{22,17}^{\left(3\right)}={\omega }_{12,9}^{\left(4\right)},{\omega }_{22,18}^{\left(3\right)}={\omega }_{12,10}^{\left(4\right)},{\omega }_{22,21}^{\left(3\right)}={\omega }_{12,11}^{\left(4\right)},{\omega }_{22,22}^{\left(3\right)}={\omega }_{12,12}^{\left(4\right)},\hfill \end{array}\right.\end{array}$$

(51)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{1,22}^{\left(3\right)}={\omega }_{1,12}^{\left(4\right)},{\omega }_{2,22}^{\left(3\right)}={\omega }_{2,12}^{\left(4\right)},{\omega }_{5,22}^{\left(3\right)}={\omega }_{3,12}^{\left(4\right)},{\omega }_{6,22}^{\left(3\right)}={\omega }_{4,12}^{\left(4\right)},{\omega }_{9,22}^{\left(3\right)}={\omega }_{5,12}^{\left(4\right)},{\omega }_{10,22}^{\left(3\right)}={\omega }_{6,12}^{\left(4\right)},\hfill \\ {\omega }_{13,22}^{\left(3\right)}={\omega }_{7,12}^{\left(4\right)},{\omega }_{14,22}^{\left(3\right)}={\omega }_{8,12}^{\left(4\right)},{\omega }_{17,22}^{\left(3\right)}={\omega }_{9,12}^{\left(4\right)},{\omega }_{18,22}^{\left(3\right)}={\omega }_{10,12}^{\left(4\right)},{\omega }_{21,22}^{\left(3\right)}={\omega }_{11,12}^{\left(4\right)},{\omega }_{22,22}^{\left(3\right)}={\omega }_{12,12}^{\left(4\right)},\hfill \end{array}\right.\end{array}$$

(52)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{21,2}^{\left(3\right)}={\omega }_{11,2}^{\left(4\right)},{\omega }_{21,6}^{\left(3\right)}={\omega }_{11,4}^{\left(4\right)},{\omega }_{21,10}^{\left(3\right)}={\omega }_{11,6}^{\left(4\right)},{\omega }_{21,14}^{\left(3\right)}={\omega }_{11,8}^{\left(4\right)},{\omega }_{21,18}^{\left(3\right)}={\omega }_{11,10}^{\left(4\right)},{\omega }_{21,22}^{\left(3\right)}={\omega }_{11,12}^{\left(4\right)},\hfill \\ {\omega }_{22,2}^{\left(3\right)}={\omega }_{12,2}^{\left(4\right)},{\omega }_{22,6}^{\left(3\right)}={\omega }_{12,4}^{\left(4\right)},{\omega }_{22,10}^{\left(3\right)}={\omega }_{12,6}^{\left(4\right)},{\omega }_{22,14}^{\left(3\right)}={\omega }_{12,8}^{\left(4\right)},{\omega }_{22,18}^{\left(3\right)}={\omega }_{12,10}^{\left(4\right)},{\omega }_{22,22}^{\left(3\right)}={\omega }_{12,12}^{\left(4\right)},\hfill \end{array}\right.\end{array}$$

(53)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{2,21}^{\left(3\right)}={\omega }_{2,11}^{\left(4\right)},{\omega }_{6,21}^{\left(3\right)}={\omega }_{4,11}^{\left(4\right)},{\omega }_{10,21}^{\left(3\right)}={\omega }_{6,11}^{\left(4\right)},{\omega }_{14,21}^{\left(3\right)}={\omega }_{8,11}^{\left(4\right)},{\omega }_{18,21}^{\left(3\right)}={\omega }_{10,11}^{\left(4\right)},{\omega }_{22,21}^{\left(3\right)}={\omega }_{12,11}^{\left(4\right)},\hfill \\ {\omega }_{2,22}^{\left(3\right)}={\omega }_{2,12}^{\left(4\right)},{\omega }_{6,22}^{\left(3\right)}={\omega }_{4,12}^{\left(4\right)},{\omega }_{10,22}^{\left(3\right)}={\omega }_{6,12}^{\left(4\right)},{\omega }_{14,22}^{\left(3\right)}={\omega }_{8,12}^{\left(4\right)},{\omega }_{18,22}^{\left(3\right)}={\omega }_{10,12}^{\left(4\right)},{\omega }_{22,22}^{\left(3\right)}={\omega }_{12,12}^{\left(4\right)},\hfill \end{array}\right.\end{array}$$

(54)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{12,1}^{\left(1\right)}+{\omega }_{81}^{\left(3\right)}={\omega }_{10,1}^{\left(2\right)},{\omega }_{12,2}^{\left(1\right)}+{\omega }_{82}^{\left(3\right)}={\omega }_{10,2}^{\left(2\right)},{\omega }_{12,3}^{\left(1\right)}+{\omega }_{83}^{\left(3\right)}={\omega }_{10,3}^{\left(2\right)},{\omega }_{12,4}^{\left(1\right)}+{\omega }_{84}^{\left(3\right)}={\omega }_{10,4}^{\left(2\right)},\hfill \\ {\omega }_{12,9}^{\left(1\right)}+{\omega }_{85}^{\left(3\right)}={\omega }_{10,7}^{\left(2\right)},{\omega }_{12,10}^{\left(1\right)}+{\omega }_{86}^{\left(3\right)}={\omega }_{10,8}^{\left(2\right)},{\omega }_{12,11}^{\left(1\right)}+{\omega }_{87}^{\left(3\right)}={\omega }_{10,9}^{\left(2\right)},{\omega }_{12,12}^{\left(1\right)}+{\omega }_{88}^{\left(3\right)}={\omega }_{10,10}^{\left(2\right)},\hfill \end{array}\right.\end{array}$$

(55)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{1,12}^{\left(1\right)}+{\omega }_{18}^{\left(3\right)}={\omega }_{1,10}^{\left(2\right)},{\omega }_{2,12}^{\left(1\right)}+{\omega }_{28}^{\left(3\right)}={\omega }_{2,10}^{\left(2\right)},{\omega }_{3,12}^{\left(1\right)}+{\omega }_{38}^{\left(3\right)}={\omega }_{3,10}^{\left(2\right)},{\omega }_{4,12}^{\left(1\right)}+{\omega }_{48}^{\left(3\right)}={\omega }_{4,10}^{\left(2\right)},\hfill \\ {\omega }_{9,12}^{\left(1\right)}+{\omega }_{58}^{\left(3\right)}={\omega }_{7,10}^{\left(2\right)},{\omega }_{10,12}^{\left(1\right)}+{\omega }_{68}^{\left(3\right)}={\omega }_{8,10}^{\left(2\right)},{\omega }_{11,12}^{\left(1\right)}+{\omega }_{78}^{\left(3\right)}={\omega }_{9,10}^{\left(2\right)},{\omega }_{12,12}^{\left(1\right)}+{\omega }_{88}^{\left(3\right)}={\omega }_{10,10}^{\left(2\right)},\hfill \end{array}\right.\end{array}$$

(56)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{94}^{\left(1\right)}+{\omega }_{54}^{\left(3\right)}={\omega }_{74}^{\left(2\right)},{\omega }_{10,4}^{\left(1\right)}+{\omega }_{64}^{\left(3\right)}={\omega }_{84}^{\left(2\right)},{\omega }_{11,4}^{\left(1\right)}+{\omega }_{74}^{\left(3\right)}={\omega }_{94}^{\left(2\right)},{\omega }_{12,4}^{\left(1\right)}+{\omega }_{84}^{\left(3\right)}={\omega }_{10,4}^{\left(2\right)},\hfill \\ {\omega }_{9,12}^{\left(1\right)}+{\omega }_{58}^{\left(3\right)}={\omega }_{7,10}^2,{\omega }_{10,12}^{\left(1\right)}+{\omega }_{68}^{\left(3\right)}={\omega }_{8,10}^2,{\omega }_{11,12}^{\left(1\right)}+{\omega }_{78}^{\left(3\right)}={\omega }_{9,10}^2,{\omega }_{12,12}^{\left(1\right)}+{\omega }_{88}^{\left(3\right)}={\omega }_{10,10}^2,\hfill \end{array}\right.\end{array}$$

(57)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{4,9}^{\left(1\right)}+{\omega }_{45}^{\left(3\right)}={\omega }_{47}^{\left(2\right)},{\omega }_{4,10}^{\left(1\right)}+{\omega }_{46}^{\left(3\right)}={\omega }_{48}^{\left(2\right)},{\omega }_{4,11}^{\left(1\right)}+{\omega }_{47}^{\left(3\right)}={\omega }_{49}^{\left(2\right)},{\omega }_{4,12}^{\left(1\right)}+{\omega }_{48}^{\left(3\right)}={\omega }_{4,10}^{\left(2\right)},\hfill \\ {\omega }_{12,9}^{\left(1\right)}+{\omega }_{85}^{\left(3\right)}={\omega }_{10,7}^2,{\omega }_{12,10}^{\left(1\right)}+{\omega }_{86}^{\left(3\right)}={\omega }_{10,8}^2,{\omega }_{12,11}^{\left(1\right)}+{\omega }_{87}^{\left(3\right)}={\omega }_{10,9}^2,{\omega }_{12,12}^{\left(1\right)}+{\omega }_{88}^{\left(3\right)}={\omega }_{10,10}^2,\hfill \end{array}\right.\end{array}$$

(58)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{20,1}^{\left(2\right)}+{\omega }_{81}^{\left(4\right)}={\omega }_{14,1}^{\left(3\right)},{\omega }_{20,2}^{\left(2\right)}+{\omega }_{82}^{\left(4\right)}={\omega }_{14,2}^{\left(3\right)},{\omega }_{20,7}^{\left(2\right)}+{\omega }_{83}^{\left(4\right)}={\omega }_{14,5}^{\left(3\right)},{\omega }_{20,8}^{\left(2\right)}+{\omega }_{84}^{\left(4\right)}={\omega }_{14,6}^{\left(3\right)},\hfill \\ {\omega }_{20,13}^{\left(2\right)}+{\omega }_{85}^{\left(4\right)}={\omega }_{14,9}^{\left(3\right)},{\omega }_{20,14}^{\left(2\right)}+{\omega }_{86}^{\left(4\right)}={\omega }_{14,10}^{\left(3\right)},{\omega }_{20,19}^{\left(2\right)}+{\omega }_{87}^{\left(4\right)}={\omega }_{14,13}^{\left(3\right)},{\omega }_{20,20}^{\left(2\right)}+{\omega }_{88}^{\left(4\right)}={\omega }_{14,14}^{\left(3\right)},\hfill \end{array}\right.\end{array}$$

(59)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{1,20}^{\left(2\right)}+{\omega }_{18}^{\left(4\right)}={\omega }_{1,14}^{\left(3\right)},{\omega }_{2,20}^{\left(2\right)}+{\omega }_{28}^{\left(4\right)}={\omega }_{2,14}^{\left(3\right)},{\omega }_{7,20}^{\left(2\right)}+{\omega }_{38}^{\left(4\right)}={\omega }_{5,14}^{\left(3\right)},{\omega }_{8,20}^{\left(2\right)}+{\omega }_{48}^{\left(4\right)}={\omega }_{6,14}^{\left(3\right)},\hfill \\ {\omega }_{13,20}^{\left(2\right)}+{\omega }_{58}^{\left(4\right)}={\omega }_{9,14}^{\left(3\right)},{\omega }_{14,20}^{\left(2\right)}+{\omega }_{68}^{\left(4\right)}={\omega }_{10,14}^{\left(3\right)},{\omega }_{19,20}^{\left(2\right)}+{\omega }_{78}^{\left(4\right)}={\omega }_{13,14}^{\left(3\right)},{\omega }_{20,20}^{\left(2\right)}+{\omega }_{88}^{\left(4\right)}={\omega }_{14,14}^{\left(3\right)},\hfill \end{array}\right.\end{array}$$

(60)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{19,2}^{\left(2\right)}+{\omega }_{72}^{\left(4\right)}={\omega }_{13,2}^{\left(3\right)},{\omega }_{20,2}^{\left(2\right)}+{\omega }_{82}^{\left(4\right)}={\omega }_{14,2}^{\left(3\right)},{\omega }_{19,8}^{\left(2\right)}+{\omega }_{74}^{\left(4\right)}={\omega }_{13,6}^{\left(3\right)},{\omega }_{20,8}^{\left(2\right)}+{\omega }_{84}^{\left(4\right)}={\omega }_{14,6}^{\left(3\right)},\hfill \\ {\omega }_{19,14}^{\left(2\right)}+{\omega }_{76}^{\left(4\right)}={\omega }_{13,10}^{\left(3\right)},{\omega }_{20,14}^{\left(2\right)}+{\omega }_{86}^{\left(4\right)}={\omega }_{14,10}^{\left(3\right)},{\omega }_{19,20}^{\left(2\right)}+{\omega }_{78}^{\left(4\right)}={\omega }_{13,14}^{\left(3\right)},{\omega }_{20,20}^{\left(2\right)}+{\omega }_{88}^{\left(4\right)}={\omega }_{14,14}^{\left(3\right)},\hfill \end{array}\right.\end{array}$$

(61)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{2,19}^{\left(2\right)}+{\omega }_{27}^{\left(4\right)}={\omega }_{2,13}^{\left(3\right)},{\omega }_{2,20}^{\left(2\right)}+{\omega }_{28}^{\left(4\right)}={\omega }_{2,14}^{\left(3\right)},{\omega }_{8,19}^{\left(2\right)}+{\omega }_{47}^{\left(4\right)}={\omega }_{6,13}^{\left(3\right)},{\omega }_{8,20}^{\left(2\right)}+{\omega }_{48}^{\left(4\right)}={\omega }_{6,14}^{\left(3\right)},\hfill \\ {\omega }_{14,19}^{\left(2\right)}+{\omega }_{67}^{\left(4\right)}={\omega }_{10,13}^{\left(3\right)},{\omega }_{14,20}^{\left(2\right)}+{\omega }_{68}^{\left(4\right)}={\omega }_{10,14}^{\left(3\right)},{\omega }_{20,19}^{\left(2\right)}+{\omega }_{87}^{\left(4\right)}={\omega }_{13,14}^{\left(3\right)},{\omega }_{20,20}^{\left(2\right)}+{\omega }_{88}^{\left(4\right)}={\omega }_{14,14}^{\left(3\right)},\hfill \end{array}\right.\end{array}$$

(62)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{10,1}^{\left(1\right)}+{\omega }_{61}^{\left(3\right)}={\omega }_{81}^{\left(2\right)}+{\omega }_{41}^{\left(4\right)},{\omega }_{10,2}^{\left(1\right)}+{\omega }_{62}^{\left(3\right)}={\omega }_{82}^{\left(2\right)}+{\omega }_{42}^{\left(4\right)},\hfill \\ {\omega }_{10,9}^{\left(1\right)}+{\omega }_{65}^{\left(3\right)}={\omega }_{87}^{\left(2\right)}+{\omega }_{43}^{\left(4\right)},{\omega }_{10,10}^{\left(1\right)}+{\omega }_{66}^{\left(3\right)}={\omega }_{88}^{\left(2\right)}+{\omega }_{44}^{\left(4\right)},\hfill \end{array}\right.\end{array}$$

(63)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{1,10}^{\left(1\right)}+{\omega }_{16}^{\left(3\right)}={\omega }_{18}^{\left(2\right)}+{\omega }_{14}^{\left(4\right)},{\omega }_{2,10}^{\left(1\right)}+{\omega }_{26}^{\left(3\right)}={\omega }_{28}^{\left(2\right)}+{\omega }_{24}^{\left(4\right)},\hfill \\ {\omega }_{9,10}^{\left(1\right)}+{\omega }_{56}^{\left(3\right)}={\omega }_{78}^{\left(2\right)}+{\omega }_{34}^{\left(4\right)},{\omega }_{10,10}^{\left(1\right)}+{\omega }_{66}^{\left(3\right)}={\omega }_{88}^{\left(2\right)}+{\omega }_{44}^{\left(4\right)},\hfill \end{array}\right.\end{array}$$

(64)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{92}^{\left(1\right)}+{\omega }_{52}^{\left(3\right)}={\omega }_{72}^{\left(2\right)}+{\omega }_{32}^{\left(4\right)},{\omega }_{10,2}^{\left(1\right)}+{\omega }_{62}^{\left(3\right)}={\omega }_{82}^{\left(2\right)}+{\omega }_{42}^{\left(4\right)},\hfill \\ {\omega }_{9,10}^{\left(1\right)}+{\omega }_{56}^{\left(3\right)}={\omega }_{78}^{\left(2\right)}+{\omega }_{34}^{\left(4\right)},{\omega }_{10,10}^{\left(1\right)}+{\omega }_{66}^{\left(3\right)}={\omega }_{88}^{\left(2\right)}+{\omega }_{44}^{\left(4\right)},\hfill \end{array}\right.\end{array}$$

(65)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\omega }_{29}^{\left(1\right)}+{\omega }_{25}^{\left(3\right)}={\omega }_{27}^{\left(2\right)}+{\omega }_{23}^{\left(4\right)},{\omega }_{2,10}^{\left(1\right)}+{\omega }_{26}^{\left(3\right)}={\omega }_{28}^{\left(2\right)}+{\omega }_{24}^{\left(4\right)},\hfill \\ {\omega }_{10,9}^{\left(1\right)}+{\omega }_{65}^{\left(3\right)}={\omega }_{87}^{\left(2\right)}+{\omega }_{43}^{\left(4\right)},{\omega }_{10,10}^{\left(1\right)}+{\omega }_{66}^{\left(3\right)}={\omega }_{88}^{\left(2\right)}+{\omega }_{44}^{\left(4\right)}.\hfill \end{array}\right.\end{array}$$

(66)

 Proof. 

(1) ⇒ (2): Suppose that $(X_1^{\prime },X_2^{\prime },X_3^{\prime },X_4^{\prime },X_5^{\prime })$ is a solution to system (1), that is,

$$\begin{array}{c}\hfill A_i{X}_i^{\prime }{C}_i+B_i{X}_{i+1}^{\prime }{D}_i={\Omega }_i,i=1,2,3,4,\end{array}$$

we can employ elementary matrix operations to show that the rank equalities (12)–(39) hold.

(2) ⇒ (3):

$$\begin{array}{c}\hfill \left(12\right)\iff r\left(S_{a_i}\widehat{{\Omega }_i}{S}_{b_i}\right)=r\left(S_{a_i}{S}_{b_i}\right)\Rightarrow \left\{\begin{array}{c}i=1,\left(\begin{array}{cccc}{\omega }_{16,1}^{\left(1\right)}& {\omega }_{16,2}^{\left(1\right)}& \cdots & {\omega }_{16,16}^{\left(1\right)}\end{array}\right)=0,\hfill \\ i=2,\left(\begin{array}{cccc}{\omega }_{24,1}^{\left(2\right)}& {\omega }_{24,2}^{\left(2\right)}& \cdots & {\omega }_{24,24}^{\left(2\right)}\end{array}\right)=0,\hfill \\ i=3,\left(\begin{array}{cccc}{\omega }_{24,1}^{\left(3\right)}& {\omega }_{24,2}^{\left(3\right)}& \cdots & {\omega }_{24,24}^{\left(3\right)}\end{array}\right)=0,\hfill \\ i=4,\left(\begin{array}{cccc}{\omega }_{16,1}^{\left(4\right)}& {\omega }_{16,2}^{\left(4\right)}& \cdots & {\omega }_{16,16}^{\left(4\right)}\end{array}\right)=0.\hfill \end{array}\right.\end{array}$$

Similarly, we have: (13) ⇒ (40), (14) ⇒ (41), (15) ⇒ (42),

(16) ⇒ (43) with (40) − (42), (17) ⇒ (44) with (40) − (42),

(18) ⇒ (45) with (40) − (42), (19) ⇒ (46) with (40) − (42),

(20) ⇒ (48) with (40) − (42), (22) ⇒ (49) with (40) − (42),

(23) ⇒ (50) with (40) − (42), (24) ⇒ (51) with (40) − (42),

(25) ⇒ (52) with (40) − (42), (26) ⇒ (53) with (40) − (42),

(27) ⇒ (54) with (40) − (42), (28) ⇒ (55) with (40) − (50),

(29) ⇒ (56) with (40) − (50), (30) ⇒ (57) with (40) − (50),

(31) ⇒ (58) with (40) − (50),

(32) ⇒ (59) with (40) − (42) and (47) − (54),

(33) ⇒ (60) with (40) − (42) and (47) − (54),

(34) ⇒ (61) with (40) − (42) and (47) − (54),

(35) ⇒ (62) with (40) − (42) and (47) − (54),

(36) ⇒ (63) with (40) − (62), (37) ⇒ (64) with (40) − (62),

(38) ⇒ (65) with (40) − (62), (39) ⇒ (66) with (40) − (62).

(3) ⇔ (1): By (8), (9), (10) and (11), system (1) is consistent if and only if (40)–(66) hold.    □

By utilizing the simultaneous decomposition, we give out some necessary and sufficient conditions for system (1) to be solvable. However, it is hard to verify the conditions (40)–(66) because the amount of them is huge. It is easy to check conditions (12)–(39). In terms of conditions (40)–(66), we put more emphasis on their mutual verification with (12)–(39). In addition, by making use of the decomposition, we can obtain some useful properties related to the general solution. We refer the readers to [5].

3. The General Solution to System (1)

In this section, we detail the general solution to system (1) by using the partitioned matrix, and Algorithm 1 which clearly illustrate the steps to obtain the general solution to system (1) is set up.

 Theorem 2. 

If (12)–(39) or (40)–(66) hold, then $X_j=Q_j\widehat{X_j}{T}_j$ are the general solution to system (1), where $j=1,2,3,4,5$. $\widehat{X_j}$ are listed as follows:

$$\begin{array}{c}\hfill \widehat{X_1}=(Y_1^{\left(1\right)},Y_2^{\left(1\right)}),\end{array}$$

(67)

where

$$\begin{array}{c}\hfill Y_1^{\left(1\right)}=\left(\begin{array}{ccc}{\omega }_{11}^{\left(1\right)}-{\omega }_{11}^{\left(2\right)}+{\omega }_{11}^{\left(3\right)}-{\omega }_{11}^{\left(4\right)}+X_{11}^{\left(5\right)}& {\omega }_{12}^{\left(1\right)}-{\omega }_{12}^{\left(2\right)}+{\omega }_{12}^{\left(3\right)}-{\omega }_{12}^{\left(4\right)}& {\omega }_{13}^{\left(1\right)}-{\omega }_{13}^{\left(2\right)}+{\omega }_{13}^{\left(3\right)}-X_{13}^{\left(4\right)}\\ {\omega }_{21}^{\left(1\right)}-{\omega }_{21}^{\left(2\right)}+{\omega }_{21}^{\left(3\right)}-{\omega }_{21}^{\left(4\right)}& {\omega }_{22}^{\left(1\right)}-{\omega }_{22}^{\left(2\right)}+{\omega }_{22}^{\left(3\right)}-{\omega }_{22}^{\left(4\right)}& {\omega }_{23}^{\left(1\right)}-{\omega }_{23}^{\left(2\right)}+{\omega }_{23}^{\left(3\right)}-X_{23}^{\left(4\right)}\\ {\omega }_{31}^{\left(1\right)}-{\omega }_{31}^{\left(2\right)}+{\omega }_{31}^{\left(3\right)}-X_{31}^{\left(4\right)}& {\omega }_{32}^{\left(1\right)}-{\omega }_{32}^{\left(2\right)}+{\omega }_{32}^{\left(3\right)}-X_{32}^{\left(4\right)}& {\omega }_{33}^{\left(1\right)}-{\omega }_{33}^{\left(2\right)}+{\omega }_{33}^{\left(3\right)}-X_{33}^{\left(4\right)}\\ {\omega }_{41}^{\left(1\right)}-{\omega }_{41}^{\left(2\right)}+{\omega }_{41}^{\left(3\right)}& {\omega }_{42}^{\left(1\right)}-{\omega }_{42}^{\left(2\right)}+{\omega }_{42}^{\left(3\right)}& {\omega }_{43}^{\left(1\right)}-{\omega }_{43}^{\left(2\right)}+{\omega }_{43}^{\left(3\right)}\\ {\omega }_{51}^{\left(1\right)}-{\omega }_{51}^{\left(2\right)}+X_{51}^{\left(3\right)}& {\omega }_{52}^{\left(1\right)}-{\omega }_{52}^{\left(2\right)}+X_{52}^{\left(3\right)}& {\omega }_{53}^{\left(1\right)}-{\omega }_{53}^{\left(2\right)}+X_{53}^{\left(3\right)}\\ {\omega }_{61}^{\left(1\right)}-{\omega }_{61}^{\left(2\right)}& {\omega }_{62}^{\left(1\right)}-{\omega }_{62}^{\left(2\right)}& {\omega }_{63}^{\left(1\right)}-{\omega }_{63}^{\left(2\right)}\\ {\omega }_{71}^{\left(1\right)}-X_{71}^{\left(2\right)}& {\omega }_{72}^{\left(1\right)}-X_{72}^{\left(2\right)}& {\omega }_{73}^{\left(1\right)}-X_{73}^{\left(2\right)}\\ {\omega }_{81}^{\left(1\right)}& {\omega }_{82}^{\left(1\right)}& {\omega }_{83}^{\left(1\right)}\\ X_{91}^{\left(1\right)}& X_{92}^{\left(1\right)}& X_{93}^{\left(1\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Y_2^{\left(1\right)}=\left(\begin{array}{cccccc}{\omega }_{14}^{\left(1\right)}-{\omega }_{14}^{\left(2\right)}+{\omega }_{14}^{\left(3\right)}& {\omega }_{15}^{\left(1\right)}-{\omega }_{15}^{\left(2\right)}+X_{15}^{\left(3\right)}& {\omega }_{16}^{\left(1\right)}-{\omega }_{16}^{\left(2\right)}& {\omega }_{17}^{\left(1\right)}-X_{17}^{\left(2\right)}& {\omega }_{18}^{\left(1\right)}& X_{19}^{\left(1\right)}\\ {\omega }_{24}^{\left(1\right)}-{\omega }_{24}^{\left(2\right)}+{\omega }_{24}^{\left(3\right)}& {\omega }_{25}^{\left(1\right)}-{\omega }_{25}^{\left(2\right)}+X_{25}^{\left(3\right)}& {\omega }_{26}^{\left(1\right)}-{\omega }_{26}^{\left(2\right)}& {\omega }_{27}^{\left(1\right)}-X_{27}^{\left(2\right)}& {\omega }_{28}^{\left(1\right)}& X_{29}^{\left(1\right)}\\ {\omega }_{34}^{\left(1\right)}-{\omega }_{34}^{\left(2\right)}+{\omega }_{34}^{\left(3\right)}& {\omega }_{35}^{\left(1\right)}-{\omega }_{35}^{\left(2\right)}+X_{35}^{\left(3\right)}& {\omega }_{36}^{\left(1\right)}-{\omega }_{36}^{\left(2\right)}& {\omega }_{37}^{\left(1\right)}-X_{37}^{\left(2\right)}& {\omega }_{38}^{\left(1\right)}& X_{39}^{\left(1\right)}\\ {\omega }_{44}^{\left(1\right)}-{\omega }_{44}^{\left(2\right)}+{\omega }_{44}^{\left(3\right)}& {\omega }_{45}^{\left(1\right)}-{\omega }_{45}^{\left(2\right)}+X_{45}^{\left(3\right)}& {\omega }_{46}^{\left(1\right)}-{\omega }_{46}^{\left(2\right)}& {\omega }_{47}^{\left(1\right)}-X_{47}^{\left(2\right)}& {\omega }_{48}^{\left(1\right)}& X_{49}^{\left(1\right)}\\ {\omega }_{54}^{\left(1\right)}-{\omega }_{54}^{\left(2\right)}+X_{54}^{\left(3\right)}& {\omega }_{55}^{\left(1\right)}-{\omega }_{55}^{\left(2\right)}+X_{55}^{\left(3\right)}& {\omega }_{56}^{\left(1\right)}-{\omega }_{56}^{\left(2\right)}& {\omega }_{57}^{\left(1\right)}-X_{57}^{\left(2\right)}& {\omega }_{58}^{\left(1\right)}& X_{59}^{\left(1\right)}\\ {\omega }_{64}^{\left(1\right)}-{\omega }_{64}^{\left(2\right)}& {\omega }_{65}^{\left(1\right)}-{\omega }_{65}^{\left(2\right)}& {\omega }_{66}^{\left(1\right)}-{\omega }_{66}^{\left(2\right)}& {\omega }_{67}^{\left(1\right)}-X_{67}^{\left(2\right)}& {\omega }_{68}^{\left(1\right)}& X_{69}^{\left(1\right)}\\ {\omega }_{74}^{\left(1\right)}-X_{74}^{\left(2\right)}& {\omega }_{75}^{\left(1\right)}-X_{75}^{\left(2\right)}& {\omega }_{76}^{\left(1\right)}-X_{76}^{\left(2\right)}& {\omega }_{77}^{\left(1\right)}-X_{77}^{\left(2\right)}& {\omega }_{78}^{\left(1\right)}& X_{79}^{\left(1\right)}\\ {\omega }_{84}^{\left(1\right)}& {\omega }_{85}^{\left(1\right)}& {\omega }_{86}^{\left(1\right)}& {\omega }_{87}^{\left(1\right)}& {\omega }_{88}^{\left(1\right)}& X_{89}^{\left(1\right)}\\ X_{94}^{\left(1\right)}& X_{95}^{\left(1\right)}& X_{96}^{\left(1\right)}& X_{97}^{\left(1\right)}& X_{98}^{\left(1\right)}& X_{99}^{\left(1\right)}\end{array}\right).\end{array}$$

$$\begin{array}{c}\hfill \widehat{X_2}=\left(\begin{array}{ccc}{Y}_{11}^{\left(2\right)}& Y_{12}^{\left(2\right)}& Y_{13}^{\left(2\right)}\\ Y_{21}^{\left(2\right)}& Y_{22}^{\left(2\right)}& Y_{23}^{\left(2\right)}\end{array}\right),\end{array}$$

(68)

where

$$\begin{array}{c}\hfill Y_{11}^{\left(2\right)}=\left(\begin{array}{ccccccc}{\omega }_{11}^{\left(2\right)}-{\omega }_{11}^{\left(3\right)}+{\omega }_{11}^{\left(4\right)}-X_{11}^{\left(5\right)}& {\omega }_{12}^{\left(2\right)}-{\omega }_{12}^{\left(3\right)}+{\omega }_{12}^{\left(4\right)}& {\omega }_{13}^{\left(2\right)}-{\omega }_{13}^{\left(3\right)}+X_{13}^{\left(4\right)}& {\omega }_{14}^{\left(2\right)}-{\omega }_{14}^{\left(3\right)}& {\omega }_{15}^{\left(2\right)}-X_{15}^{\left(3\right)}& {\omega }_{16}^{\left(2\right)}& X_{17}^{\left(2\right)}\\ {\omega }_{21}^{\left(2\right)}-{\omega }_{21}^{\left(3\right)}+{\omega }_{21}^{\left(4\right)}& {\omega }_{22}^{\left(2\right)}-{\omega }_{22}^{\left(3\right)}+{\omega }_{22}^{\left(4\right)}& {\omega }_{23}^{\left(2\right)}-{\omega }_{23}^{\left(3\right)}+X_{23}^{\left(4\right)}& {\omega }_{24}^{\left(2\right)}-{\omega }_{24}^{\left(3\right)}& {\omega }_{25}^{\left(2\right)}-X_{25}^{\left(3\right)}& {\omega }_{26}^{\left(2\right)}& X_{27}^{\left(2\right)}\\ {\omega }_{31}^{\left(2\right)}-{\omega }_{31}^{\left(3\right)}+X_{31}^{\left(4\right)}& {\omega }_{32}^{\left(2\right)}-{\omega }_{32}^{\left(3\right)}+X_{32}^{\left(4\right)}& {\omega }_{33}^{\left(2\right)}-{\omega }_{33}^{\left(3\right)}+X_{33}^{\left(4\right)}& {\omega }_{34}^{\left(2\right)}-{\omega }_{34}^{\left(3\right)}& {\omega }_{35}^{\left(2\right)}-X_{35}^{\left(3\right)}& {\omega }_{36}^{\left(2\right)}& X_{37}^{\left(2\right)}\\ {\omega }_{41}^{\left(2\right)}-{\omega }_{41}^{\left(3\right)}& {\omega }_{42}^{\left(2\right)}-{\omega }_{42}^{\left(3\right)}& {\omega }_{43}^{\left(2\right)}-{\omega }_{43}^{\left(3\right)}& {\omega }_{44}^{\left(2\right)}-{\omega }_{44}^{\left(3\right)}& {\omega }_{45}^{\left(2\right)}-X_{45}^{\left(3\right)}& {\omega }_{46}^{\left(2\right)}& X_{47}^{\left(2\right)}\\ {\omega }_{51}^{\left(2\right)}-X_{51}^{\left(3\right)}& {\omega }_{52}^{\left(2\right)}-X_{52}^{\left(3\right)}& {\omega }_{53}^{\left(2\right)}-X_{53}^{\left(3\right)}& {\omega }_{54}^{\left(2\right)}-X_{54}^{\left(3\right)}& {\omega }_{55}^{\left(2\right)}-X_{55}^{\left(3\right)}& {\omega }_{56}^{\left(2\right)}& X_{57}^{\left(2\right)}\\ {\omega }_{61}^{\left(2\right)}& {\omega }_{62}^{\left(2\right)}& {\omega }_{63}^{\left(2\right)}& {\omega }_{64}^{\left(2\right)}& {\omega }_{65}^{\left(2\right)}& {\omega }_{66}^{\left(2\right)}& X_{67}^{\left(2\right)}\\ X_{71}^{\left(2\right)}& X_{72}^{\left(2\right)}& X_{73}^{\left(2\right)}& X_{74}^{\left(2\right)}& X_{75}^{\left(2\right)}& X_{76}^{\left(2\right)}& X_{77}^{\left(2\right)}\\ {\omega }_{91}^{\left(1\right)}& {\omega }_{92}^{\left(1\right)}& {\omega }_{93}^{\left(1\right)}& {\omega }_{94}^{\left(1\right)}& {\omega }_{95}^{\left(1\right)}& {\omega }_{96}^{\left(1\right)}& {\omega }_{97}^{\left(1\right)}\\ {\omega }_{10,1}^{\left(1\right)}& {\omega }_{10,2}^{\left(1\right)}& {\omega }_{10,3}^{\left(1\right)}& {\omega }_{10,4}^{\left(1\right)}& {\omega }_{10,5}^{\left(1\right)}& {\omega }_{10,6}^{\left(1\right)}& {\omega }_{10,7}^{\left(1\right)}\\ {\omega }_{11,1}^{\left(1\right)}& {\omega }_{11,2}^{\left(1\right)}& {\omega }_{11,3}^{\left(1\right)}& {\omega }_{11,4}^{\left(1\right)}& {\omega }_{11,5}^{\left(1\right)}& {\omega }_{11,6}^{\left(1\right)}& {\omega }_{11,7}^{\left(1\right)}\end{array}\right),\end{array}$$

$$\begin{array}{cc}\hfill & Y_{21}^{\left(2\right)}=\hfill \\ \hfill & \left(\begin{array}{ccccccc}{\omega }_{12,1}^{\left(1\right)}& {\omega }_{12,2}^{\left(1\right)}& {\omega }_{12,3}^{\left(1\right)}& {\omega }_{12,4}^{\left(1\right)}& {\omega }_{12,5}^{\left(1\right)}& {\omega }_{12,6}^{\left(1\right)}& {\omega }_{12,7}^{\left(1\right)}\\ {\omega }_{13,1}^{\left(1\right)}& {\omega }_{13,2}^{\left(1\right)}& {\omega }_{13,3}^{\left(1\right)}& {\omega }_{13,4}^{\left(1\right)}& {\omega }_{13,5}^{\left(1\right)}& {\omega }_{13,6}^{\left(1\right)}& {\omega }_{13,7}^{\left(1\right)}\\ {\omega }_{14,1}^{\left(1\right)}& {\omega }_{14,2}^{\left(1\right)}& {\omega }_{14,3}^{\left(1\right)}& {\omega }_{14,4}^{\left(1\right)}& {\omega }_{14,5}^{\left(1\right)}& {\omega }_{14,6}^{\left(1\right)}& {\omega }_{14,7}^{\left(1\right)}\\ {\omega }_{15,1}^{\left(1\right)}& {\omega }_{15,2}^{\left(1\right)}& {\omega }_{15,3}^{\left(1\right)}& {\omega }_{15,4}^{\left(1\right)}& {\omega }_{15,5}^{\left(1\right)}& {\omega }_{15,6}^{\left(1\right)}& {\omega }_{15,7}^{\left(1\right)}\\ {\omega }_{13,1}^{\left(2\right)}-{\omega }_{91}^{\left(3\right)}+{\omega }_{51}^{\left(4\right)}-X_{31}^{\left(5\right)}& {\omega }_{13,2}^{\left(2\right)}-{\omega }_{92}^{\left(3\right)}+{\omega }_{52}^{\left(4\right)}& {\omega }_{13,3}^{\left(2\right)}-{\omega }_{93}^{\left(3\right)}+X_{73}^{\left(4\right)}& {\omega }_{13,4}^{\left(2\right)}-{\omega }_{94}^{\left(3\right)}& {\omega }_{13,5}^{\left(2\right)}-X_{11,5}^{\left(3\right)}& {\omega }_{13,6}^{\left(2\right)}& X_{15,7}^{\left(2\right)}\\ {\omega }_{14,1}^{\left(2\right)}-{\omega }_{10,1}^{\left(3\right)}+{\omega }_{61}^{\left(4\right)}& {\omega }_{14,2}^{\left(2\right)}-{\omega }_{10,2}^{\left(3\right)}+{\omega }_{62}^{\left(4\right)}& {\omega }_{14,3}^{\left(2\right)}-{\omega }_{10,3}^{\left(3\right)}+X_{83}^{\left(4\right)}& {\omega }_{14,4}^{\left(2\right)}-{\omega }_{10,4}^{\left(3\right)}& {\omega }_{14,5}^{\left(2\right)}-X_{12,5}^{\left(3\right)}& {\omega }_{14,6}^{\left(2\right)}& X_{16,7}^{\left(2\right)}\\ {\omega }_{15,1}^{\left(2\right)}-{\omega }_{11,1}^{\left(3\right)}+X_{91}^{\left(4\right)}& {\omega }_{15,2}^{\left(2\right)}-{\omega }_{11,2}^{\left(3\right)}+X_{92}^{\left(4\right)}& {\omega }_{15,3}^{\left(2\right)}-{\omega }_{11,3}^{\left(3\right)}+X_{93}^{\left(4\right)}& {\omega }_{15,4}^{\left(2\right)}-{\omega }_{11,4}^{\left(3\right)}& {\omega }_{15,5}^{\left(2\right)}-X_{13,5}^{\left(3\right)}& {\omega }_{15,6}^{\left(2\right)}& X_{17,7}^{\left(2\right)}\\ {\omega }_{16,1}^{\left(2\right)}-{\omega }_{12,1}^{\left(3\right)}& {\omega }_{16,2}^{\left(2\right)}-{\omega }_{12,2}^{\left(3\right)}& {\omega }_{16,3}^{\left(2\right)}-{\omega }_{12,3}^{\left(3\right)}& {\omega }_{16,4}^{\left(2\right)}-{\omega }_{12,4}^{\left(3\right)}& {\omega }_{16,5}^{\left(2\right)}-X_{14,5}^{\left(3\right)}& {\omega }_{16,6}^{\left(2\right)}& X_{18,7}^{\left(2\right)}\\ {\omega }_{17,1}^{\left(2\right)}-X_{15,1}^{\left(3\right)}& {\omega }_{17,2}^{\left(2\right)}-X_{15,2}^{\left(3\right)}& {\omega }_{17,3}^{\left(2\right)}-X_{15,3}^{\left(3\right)}& {\omega }_{17,4}^{\left(2\right)}-X_{15,4}^{\left(3\right)}& {\omega }_{17,5}^{\left(2\right)}-X_{15,5}^{\left(3\right)}& {\omega }_{17,6}^{\left(2\right)}& X_{19,7}^{\left(2\right)}\\ {\omega }_{18,1}^{\left(2\right)}& {\omega }_{18,2}^{\left(2\right)}& {\omega }_{18,3}^{\left(2\right)}& {\omega }_{18,4}^{\left(2\right)}& {\omega }_{18,5}^{\left(2\right)}& {\omega }_{18,6}^{\left(2\right)}& X_{20,7}^{\left(2\right)}\\ X_{21,1}^{\left(2\right)}& X_{21,2}^{\left(2\right)}& X_{21,3}^{\left(2\right)}& X_{21,4}^{\left(2\right)}& X_{21,5}^{\left(2\right)}& X_{21,6}^{\left(2\right)}& X_{21,7}^{\left(2\right)}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill Y_{12}^{\left(2\right)}=\left(\begin{array}{ccccccc}{\omega }_{19}^{\left(1\right)}& {\omega }_{1,10}^{\left(1\right)}& {\omega }_{1,11}^{\left(1\right)}& {\omega }_{1,12}^{\left(1\right)}& {\omega }_{1,13}^{\left(1\right)}& {\omega }_{1,14}^{\left(1\right)}& {\omega }_{1,15}^{\left(1\right)}\\ {\omega }_{29}^{\left(1\right)}& {\omega }_{2,10}^{\left(1\right)}& {\omega }_{2,11}^{\left(1\right)}& {\omega }_{2,12}^{\left(1\right)}& {\omega }_{2,13}^{\left(1\right)}& {\omega }_{2,14}^{\left(1\right)}& {\omega }_{2,15}^{\left(1\right)}\\ {\omega }_{39}^{\left(1\right)}& {\omega }_{3,10}^{\left(1\right)}& {\omega }_{3,11}^{\left(1\right)}& {\omega }_{3,12}^{\left(1\right)}& {\omega }_{3,13}^{\left(1\right)}& {\omega }_{3,14}^{\left(1\right)}& {\omega }_{3,15}^{\left(1\right)}\\ {\omega }_{49}^{\left(1\right)}& {\omega }_{4,10}^{\left(1\right)}& {\omega }_{4,11}^{\left(1\right)}& {\omega }_{4,12}^{\left(1\right)}& {\omega }_{4,13}^{\left(1\right)}& {\omega }_{4,14}^{\left(1\right)}& {\omega }_{4,15}^{\left(1\right)}\\ {\omega }_{59}^{\left(1\right)}& {\omega }_{5,10}^{\left(1\right)}& {\omega }_{5,11}^{\left(1\right)}& {\omega }_{5,12}^{\left(1\right)}& {\omega }_{5,13}^{\left(1\right)}& {\omega }_{5,14}^{\left(1\right)}& {\omega }_{5,15}^{\left(1\right)}\\ {\omega }_{69}^{\left(1\right)}& {\omega }_{6,10}^{\left(1\right)}& {\omega }_{6,11}^{\left(1\right)}& {\omega }_{6,12}^{\left(1\right)}& {\omega }_{6,13}^{\left(1\right)}& {\omega }_{6,14}^{\left(1\right)}& {\omega }_{6,15}^{\left(1\right)}\\ {\omega }_{79}^{\left(1\right)}& {\omega }_{7,10}^{\left(1\right)}& {\omega }_{7,11}^{\left(1\right)}& {\omega }_{7,12}^{\left(1\right)}& {\omega }_{7,13}^{\left(1\right)}& {\omega }_{7,14}^{\left(1\right)}& {\omega }_{7,15}^{\left(1\right)}\\ {\omega }_{99}^{\left(1\right)}& {\omega }_{9,10}^{\left(1\right)}& {\omega }_{9,11}^{\left(1\right)}& {\omega }_{9,12}^{\left(1\right)}& {\omega }_{9,13}^{\left(1\right)}& {\omega }_{9,14}^{\left(1\right)}& {\omega }_{9,15}^{\left(1\right)}\\ {\omega }_{10,9}^{\left(1\right)}& {\omega }_{10,10}^{\left(1\right)}& {\omega }_{10,11}^{\left(1\right)}& {\omega }_{10,12}^{\left(1\right)}& {\omega }_{10,13}^{\left(1\right)}& {\omega }_{10,14}^{\left(1\right)}& {\omega }_{10,15}^{\left(1\right)}\\ {\omega }_{11,9}^{\left(1\right)}& {\omega }_{11,10}^{\left(1\right)}& {\omega }_{11,11}^{\left(1\right)}& {\omega }_{11,12}^{\left(1\right)}& {\omega }_{11,13}^{\left(1\right)}& {\omega }_{11,14}^{\left(1\right)}& {\omega }_{11,15}^{\left(1\right)}\end{array}\right),\end{array}$$

$$\begin{array}{cc}\hfill & Y_{22}^{\left(2\right)}=\hfill \\ \hfill & \left(\begin{array}{ccccccc}{\omega }_{12,9}^{\left(1\right)}& {\omega }_{12,10}^{\left(1\right)}& {\omega }_{12,11}^{\left(1\right)}& {\omega }_{12,12}^{\left(1\right)}& {\omega }_{12,13}^{\left(1\right)}& {\omega }_{12,14}^{\left(1\right)}& {\omega }_{12,15}^{\left(1\right)}\\ {\omega }_{13,9}^{\left(1\right)}& {\omega }_{13,10}^{\left(1\right)}& {\omega }_{13,11}^{\left(1\right)}& {\omega }_{13,12}^{\left(1\right)}& {\omega }_{13,13}^{\left(1\right)}& {\omega }_{13,14}^{\left(1\right)}& {\omega }_{13,15}^{\left(1\right)}\\ {\omega }_{14,9}^{\left(1\right)}& {\omega }_{14,10}^{\left(1\right)}& {\omega }_{14,11}^{\left(1\right)}& {\omega }_{14,12}^{\left(1\right)}& {\omega }_{14,13}^{\left(1\right)}& {\omega }_{14,14}^{\left(1\right)}& {\omega }_{14,15}^{\left(1\right)}\\ {\omega }_{15,9}^{\left(1\right)}& {\omega }_{15,10}^{\left(1\right)}& {\omega }_{15,11}^{\left(1\right)}& {\omega }_{15,12}^{\left(1\right)}& {\omega }_{15,13}^{\left(1\right)}& {\omega }_{15,14}^{\left(1\right)}& {\omega }_{15,15}^{\left(1\right)}\\ {\omega }_{13,7}^{\left(2\right)}-{\omega }_{95}^{\left(3\right)}+{\omega }_{53}^{\left(4\right)}-X_{32}^{\left(5\right)}& {\omega }_{13,8}^{\left(2\right)}-{\omega }_{96}^{\left(3\right)}+{\omega }_{54}^{\left(4\right)}& {\omega }_{13,9}^{\left(2\right)}-{\omega }_{97}^{\left(3\right)}+X_{76}^{\left(4\right)}& {\omega }_{13,10}^{\left(2\right)}-{\omega }_{98}^{\left(3\right)}& {\omega }_{13,11}^{\left(2\right)}-X_{11,10}^{\left(3\right)}& {\omega }_{13,12}^{\left(3\right)}& X_{15,14}^{\left(2\right)}\\ {\omega }_{14,7}^{\left(2\right)}-{\omega }_{10,5}^{\left(3\right)}+{\omega }_{63}^{\left(4\right)}& {\omega }_{14,8}^{\left(2\right)}-{\omega }_{10,6}^{\left(3\right)}+{\omega }_{64}^{\left(4\right)}& {\omega }_{14,9}^{\left(2\right)}-{\omega }_{10,7}^{\left(3\right)}+X_{86}^{\left(4\right)}& {\omega }_{14,10}^{\left(2\right)}-{\omega }_{10,8}^{\left(3\right)}& {\omega }_{14,11}^{\left(2\right)}-X_{12,10}^{\left(3\right)}& {\omega }_{14,12}^{\left(3\right)}& X_{16,14}^{\left(2\right)}\\ {\omega }_{15,7}^{\left(2\right)}-{\omega }_{11,5}^{\left(3\right)}+X_{94}^{\left(4\right)}& {\omega }_{15,8}^{\left(2\right)}-{\omega }_{11,6}^{\left(3\right)}+X_{95}^{\left(4\right)}& {\omega }_{15,9}^{\left(2\right)}-{\omega }_{11,7}^{\left(3\right)}+X_{96}^{\left(4\right)}& {\omega }_{15,10}^{\left(2\right)}-{\omega }_{11,8}^{\left(3\right)}& {\omega }_{15,11}^{\left(2\right)}-X_{13,10}^{\left(3\right)}& {\omega }_{15,12}^{\left(3\right)}& X_{17,14}^{\left(2\right)}\\ {\omega }_{16,7}^{\left(2\right)}-{\omega }_{12,5}^{\left(3\right)}& {\omega }_{16,8}^{\left(2\right)}-{\omega }_{12,6}^{\left(3\right)}& {\omega }_{16,9}^{\left(2\right)}-{\omega }_{12,7}^{\left(3\right)}& {\omega }_{16,10}^{\left(2\right)}-{\omega }_{12,8}^{\left(3\right)}& {\omega }_{16,11}^{\left(2\right)}-X_{14,10}^{\left(3\right)}& {\omega }_{16,12}^{\left(3\right)}& X_{18,14}^{\left(2\right)}\\ {\omega }_{17,7}^{\left(2\right)}-X_{15,6}^{\left(3\right)}& {\omega }_{17,8}^{\left(2\right)}-X_{15,7}^{\left(3\right)}& {\omega }_{17,9}^{\left(2\right)}-X_{15,8}^{\left(3\right)}& {\omega }_{17,10}^{\left(2\right)}-X_{15,9}^{\left(3\right)}& {\omega }_{17,11}^{\left(2\right)}-X_{15,10}^{\left(3\right)}& {\omega }_{17,12}^{\left(3\right)}& X_{19,14}^{\left(2\right)}\\ {\omega }_{18,7}^{\left(2\right)}& {\omega }_{18,8}^{\left(2\right)}& {\omega }_{18,9}^{\left(2\right)}& {\omega }_{18,10}^{\left(2\right)}& {\omega }_{18,11}^{\left(2\right)}& {\omega }_{18,12}^{\left(3\right)}& X_{20,14}^{\left(2\right)}\\ X_{21,8}^{\left(2\right)}& X_{21,9}^{\left(2\right)}& X_{21,10}^{\left(2\right)}& X_{21,11}^{\left(2\right)}& X_{21,12}^{\left(2\right)}& X_{21,13}^{\left(2\right)}& X_{21,14}^{\left(2\right)}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & Y_{13}^{\left(2\right)}=\hfill \\ \hfill & \left(\begin{array}{ccccccc}{\omega }_{1,13}^{\left(2\right)}-{\omega }_{19}^{\left(3\right)}+{\omega }_{15}^{\left(4\right)}-X_{13}^{\left(5\right)}& {\omega }_{1,14}^{\left(2\right)}-{\omega }_{1,10}^{\left(3\right)}+{\omega }_{16}^{\left(4\right)}& {\omega }_{1,15}^{\left(2\right)}-{\omega }_{1,11}^{\left(3\right)}+X_{19}^{\left(4\right)}& {\omega }_{1,16}^{\left(2\right)}-{\omega }_{1,12}^{\left(3\right)}& {\omega }_{1,17}^{\left(2\right)}-X_{1,15}^{\left(3\right)}& {\omega }_{1,18}^{\left(2\right)}& X_{1,21}^{\left(2\right)}\\ {\omega }_{2,13}^{\left(2\right)}-{\omega }_{29}^{\left(3\right)}+{\omega }_{25}^{\left(4\right)}& {\omega }_{2,14}^{\left(2\right)}-{\omega }_{2,10}^{\left(3\right)}+{\omega }_{26}^{\left(4\right)}& {\omega }_{2,15}^{\left(2\right)}-{\omega }_{2,11}^{\left(3\right)}+X_{29}^{\left(4\right)}& {\omega }_{2,16}^{\left(2\right)}-{\omega }_{2,12}^{\left(3\right)}& {\omega }_{2,17}^{\left(2\right)}-X_{2,15}^{\left(3\right)}& {\omega }_{2,18}^{\left(2\right)}& X_{2,21}^{\left(2\right)}\\ {\omega }_{3,13}^{\left(2\right)}-{\omega }_{39}^{\left(3\right)}+X_{37}^{\left(4\right)}& {\omega }_{3,14}^{\left(2\right)}-{\omega }_{3,10}^{\left(3\right)}+X_{38}^{\left(4\right)}& {\omega }_{3,15}^{\left(2\right)}-{\omega }_{3,11}^{\left(3\right)}+X_{39}^{\left(4\right)}& {\omega }_{3,16}^{\left(2\right)}-{\omega }_{3,12}^{\left(3\right)}& {\omega }_{3,17}^{\left(2\right)}-X_{3,15}^{\left(3\right)}& {\omega }_{3,18}^{\left(2\right)}& X_{3,21}^{\left(2\right)}\\ {\omega }_{4,13}^{\left(2\right)}-{\omega }_{49}^{\left(3\right)}& {\omega }_{4,14}^{\left(2\right)}-{\omega }_{4,10}^{\left(3\right)}& {\omega }_{4,15}^{\left(2\right)}-{\omega }_{4,11}^{\left(3\right)}& {\omega }_{4,16}^{\left(2\right)}-{\omega }_{4,12}^{\left(3\right)}& {\omega }_{4,17}^{\left(2\right)}-X_{4,15}^{\left(3\right)}& {\omega }_{4,18}^{\left(2\right)}& X_{4,21}^{\left(2\right)}\\ {\omega }_{5,13}^{\left(2\right)}-X_{5,11}^{\left(3\right)}& {\omega }_{5,14}^{\left(2\right)}-X_{5,12}^{\left(3\right)}& {\omega }_{5,15}^{\left(2\right)}-X_{5,13}^{\left(3\right)}& {\omega }_{5,16}^{\left(2\right)}-X_{5,14}^{\left(3\right)}& {\omega }_{5,17}^{\left(2\right)}-X_{5,15}^{\left(3\right)}& {\omega }_{5,18}^{\left(2\right)}& X_{5,21}^{\left(2\right)}\\ {\omega }_{6,13}^{\left(2\right)}& {\omega }_{6,14}^{\left(2\right)}& {\omega }_{6,15}^{\left(2\right)}& {\omega }_{6,16}^{\left(2\right)}& {\omega }_{6,17}^{\left(2\right)}& {\omega }_{6,18}^{\left(2\right)}& X_{6,21}^{\left(2\right)}\\ X_{7,15}^{\left(2\right)}& X_{7,16}^{\left(2\right)}& X_{7,17}^{\left(2\right)}& X_{7,18}^{\left(2\right)}& X_{7,19}^{\left(2\right)}& X_{7,20}^{\left(2\right)}& X_{7,21}^{\left(2\right)}\\ {\omega }_{7,13}^{\left(2\right)}-{\omega }_{59}^{\left(3\right)}+{\omega }_{35}^{\left(4\right)}-X_{23}^{\left(5\right)}& {\omega }_{7,14}^{\left(2\right)}-{\omega }_{5,10}^{\left(3\right)}+{\omega }_{36}^{\left(4\right)}& {\omega }_{7,15}^{\left(2\right)}-{\omega }_{5,11}^{\left(3\right)}+X_{49}^{\left(4\right)}& {\omega }_{7,16}^{\left(2\right)}-{\omega }_{5,12}^{\left(3\right)}& {\omega }_{7,17}^{\left(2\right)}-X_{6,15}^{\left(3\right)}& {\omega }_{7,18}^{\left(2\right)}& X_{8,21}^{\left(2\right)}\\ {\omega }_{8,13}^{\left(2\right)}-{\omega }_{69}^{\left(3\right)}+{\omega }_{45}^{\left(4\right)}& {\omega }_{8,14}^{\left(2\right)}-{\omega }_{6,10}^{\left(3\right)}+{\omega }_{46}^{\left(4\right)}& {\omega }_{8,15}^{\left(2\right)}-{\omega }_{6,11}^{\left(3\right)}+X_{59}^{\left(4\right)}& {\omega }_{8,16}^{\left(2\right)}-{\omega }_{6,12}^{\left(3\right)}& {\omega }_{8,17}^{\left(2\right)}-X_{7,15}^{\left(3\right)}& {\omega }_{8,18}^{\left(2\right)}& X_{9,21}^{\left(2\right)}\\ {\omega }_{9,13}^{\left(2\right)}-{\omega }_{79}^{\left(3\right)}+X_{67}^{\left(4\right)}& {\omega }_{9,14}^{\left(2\right)}-{\omega }_{7,10}^{\left(3\right)}+X_{68}^{\left(4\right)}& {\omega }_{9,15}^{\left(2\right)}-{\omega }_{7,11}^{\left(3\right)}+X_{69}^{\left(4\right)}& {\omega }_{9,16}^{\left(2\right)}-{\omega }_{7,12}^{\left(3\right)}& {\omega }_{9,17}^{\left(2\right)}-X_{8,15}^{\left(3\right)}& {\omega }_{9,18}^{\left(2\right)}& X_{10,21}^{\left(2\right)}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & Y_{23}^{\left(2\right)}=\hfill \\ \hfill & \left(\begin{array}{ccccccc}{\omega }_{10,13}^{\left(2\right)}-{\omega }_{89}^{\left(3\right)}& {\omega }_{10,14}^{\left(2\right)}-{\omega }_{8,10}^{\left(3\right)}& {\omega }_{10,15}^{\left(2\right)}-{\omega }_{8,11}^{\left(3\right)}& {\omega }_{10,16}^{\left(2\right)}-{\omega }_{8,12}^{\left(3\right)}& {\omega }_{10,17}^{\left(2\right)}-X_{9,15}^{\left(3\right)}& {\omega }_{10,18}^{\left(2\right)}& X_{11,21}^{\left(2\right)}\\ {\omega }_{11,13}^{\left(2\right)}-X_{10,11}^{\left(3\right)}& {\omega }_{11,14}^{\left(2\right)}-X_{10,12}^{\left(3\right)}& {\omega }_{11,15}^{\left(2\right)}-X_{10,13}^{\left(3\right)}& {\omega }_{11,16}^{\left(2\right)}-X_{10,14}^{\left(3\right)}& {\omega }_{11,17}^{\left(2\right)}-X_{10,15}^{\left(3\right)}& {\omega }_{11,18}^{\left(2\right)}& X_{12,21}^{\left(2\right)}\\ {\omega }_{12,13}^{\left(2\right)}& {\omega }_{12,14}^{\left(2\right)}& {\omega }_{12,15}^{\left(2\right)}& {\omega }_{12,16}^{\left(2\right)}& {\omega }_{12,17}^{\left(2\right)}& {\omega }_{12,18}^{\left(2\right)}& X_{13,21}^{\left(2\right)}\\ X_{14,15}^{\left(2\right)}& X_{14,16}^{\left(2\right)}& X_{14,17}^{\left(2\right)}& X_{14,18}^{\left(2\right)}& X_{14,19}^{\left(2\right)}& X_{14,20}^{\left(2\right)}& X_{14,21}^{\left(2\right)}\\ {\omega }_{13,13}^{\left(2\right)}-{\omega }_{99}^{\left(3\right)}+{\omega }_{55}^{\left(4\right)}-X_{33}^{\left(5\right)}& {\omega }_{13,14}^{\left(2\right)}-{\omega }_{9,10}^{\left(3\right)}+{\omega }_{56}^{\left(4\right)}& {\omega }_{13,15}^{\left(2\right)}-{\omega }_{9,11}^{\left(3\right)}+X_{79}^{\left(4\right)}& {\omega }_{13,16}^{\left(2\right)}-{\omega }_{9,12}^{\left(3\right)}& {\omega }_{13,17}^{\left(2\right)}-X_{11,15}^{\left(3\right)}& {\omega }_{13,18}^{\left(2\right)}& X_{15,21}^{\left(2\right)}\\ {\omega }_{14,13}^{\left(2\right)}-{\omega }_{10,9}^{\left(3\right)}+{\omega }_{65}^{\left(4\right)}& {\omega }_{14,14}^{\left(2\right)}-{\omega }_{10,10}^{\left(3\right)}+{\omega }_{66}^{\left(4\right)}& {\omega }_{14,15}^{\left(2\right)}-{\omega }_{10,11}^{\left(3\right)}+X_{89}^{\left(4\right)}& {\omega }_{14,16}^{\left(2\right)}-{\omega }_{10,12}^{\left(3\right)}& {\omega }_{14,17}^{\left(2\right)}-X_{12,15}^{\left(3\right)}& {\omega }_{14,18}^{\left(2\right)}& X_{16,21}^{\left(2\right)}\\ {\omega }_{15,13}^{\left(2\right)}-{\omega }_{11,9}^{\left(3\right)}+X_{97}^{\left(4\right)}& {\omega }_{15,14}^{\left(2\right)}-{\omega }_{11,10}^{\left(3\right)}+X_{98}^{\left(4\right)}& {\omega }_{15,15}^{\left(2\right)}-{\omega }_{11,11}^{\left(3\right)}+X_{99}^{\left(4\right)}& {\omega }_{15,16}^{\left(2\right)}-{\omega }_{11,12}^{\left(3\right)}& {\omega }_{15,17}^{\left(2\right)}-X_{13,15}^{\left(3\right)}& {\omega }_{15,18}^{\left(2\right)}& X_{17,21}^{\left(2\right)}\\ {\omega }_{16,13}^{\left(2\right)}-{\omega }_{12,9}^{\left(3\right)}& {\omega }_{16,14}^{\left(2\right)}-{\omega }_{12,10}^{\left(3\right)}& {\omega }_{16,15}^{\left(2\right)}-{\omega }_{12,11}^{\left(3\right)}& {\omega }_{16,16}^{\left(2\right)}-{\omega }_{12,12}^{\left(3\right)}& {\omega }_{16,17}^{\left(2\right)}-X_{14,15}^{\left(3\right)}& {\omega }_{16,18}^{\left(2\right)}& X_{18,21}^{\left(2\right)}\\ {\omega }_{17,13}^{\left(2\right)}-X_{15,11}^{\left(3\right)}& {\omega }_{17,14}^{\left(2\right)}-X_{15,12}^{\left(3\right)}& {\omega }_{17,15}^{\left(2\right)}-X_{15,13}^{\left(3\right)}& {\omega }_{17,16}^{\left(2\right)}-X_{15,14}^{\left(3\right)}& {\omega }_{17,17}^{\left(2\right)}-X_{15,15}^{\left(3\right)}& {\omega }_{17,18}^{\left(2\right)}& X_{19,21}^{\left(2\right)}\\ {\omega }_{18,13}^{\left(2\right)}& {\omega }_{18,14}^{\left(2\right)}& {\omega }_{18,15}^{\left(2\right)}& {\omega }_{18,16}^{\left(2\right)}& {\omega }_{18,17}^{\left(2\right)}& {\omega }_{18,18}^{\left(2\right)}& X_{20,21}^{\left(2\right)}\\ X_{21,15}^{\left(2\right)}& X_{21,16}^{\left(2\right)}& X_{21,17}^{\left(2\right)}& X_{21,18}^{\left(2\right)}& X_{21,19}^{\left(2\right)}& X_{21,20}^{\left(2\right)}& X_{21,21}^{\left(2\right)}\end{array}\right).\hfill \end{array}$$

$$\begin{array}{c}\hfill \widehat{X_3}=\left(\begin{array}{cccc}{Y}_{11}^{\left(3\right)}& Y_{12}^{\left(3\right)}& Y_{13}^{\left(3\right)}& Y_{14}^{\left(3\right)}\\ Y_{21}^{\left(3\right)}& Y_{22}^{\left(3\right)}& Y_{23}^{\left(3\right)}& Y_{24}^{\left(3\right)}\end{array}\right),\end{array}$$

(69)

where

$$\begin{array}{c}\hfill Y_{11}^{\left(3\right)}=\left(\begin{array}{cccccc}{\omega }_{11}^{\left(3\right)}-{\omega }_{11}^{\left(4\right)}+X_{11}^{\left(5\right)}& {\omega }_{12}^{\left(3\right)}-{\omega }_{12}^{\left(4\right)}& {\omega }_{13}^{\left(3\right)}-X_{13}^{\left(4\right)}& {\omega }_{14}^{\left(3\right)}& X_{15}^{\left(3\right)}& {\omega }_{17}^{\left(2\right)}-{\omega }_{19}^{\left(1\right)}\\ {\omega }_{21}^{\left(3\right)}-{\omega }_{21}^{\left(4\right)}& {\omega }_{22}^{\left(3\right)}-{\omega }_{22}^{\left(4\right)}& {\omega }_{23}^{\left(3\right)}-X_{23}^{\left(4\right)}& {\omega }_{24}^{\left(3\right)}& X_{25}^{\left(3\right)}& {\omega }_{27}^{\left(2\right)}-{\omega }_{29}^{\left(1\right)}\\ {\omega }_{31}^{\left(3\right)}-X_{31}^{\left(4\right)}& {\omega }_{32}^{\left(3\right)}-X_{32}^{\left(4\right)}& {\omega }_{33}^{\left(3\right)}-X_{33}^{\left(4\right)}& {\omega }_{34}^{\left(3\right)}& X_{35}^{\left(3\right)}& {\omega }_{37}^{\left(2\right)}-{\omega }_{39}^{\left(1\right)}\\ {\omega }_{41}^{\left(3\right)}& {\omega }_{42}^{\left(3\right)}& {\omega }_{43}^{\left(3\right)}& {\omega }_{44}^{\left(3\right)}& X_{45}^{\left(3\right)}& {\omega }_{45}^{\left(3\right)}\\ X_{51}^{\left(3\right)}& X_{52}^{\left(3\right)}& X_{53}^{\left(3\right)}& X_{54}^{\left(3\right)}& X_{55}^{\left(3\right)}& {\omega }_{57}^{\left(2\right)}-{\omega }_{59}^{\left(1\right)}\\ {\omega }_{71}^{\left(2\right)}-{\omega }_{91}^{\left(1\right)}& {\omega }_{72}^{\left(2\right)}-{\omega }_{92}^{\left(1\right)}& {\omega }_{73}^{\left(2\right)}-{\omega }_{93}^{\left(1\right)}& {\omega }_{54}^{\left(3\right)}& {\omega }_{75}^{\left(2\right)}-{\omega }_{95}^{\left(1\right)}& {\omega }_{77}^{\left(2\right)}-{\omega }_{99}^{\left(1\right)}\\ {\omega }_{81}^{\left(2\right)}-{\omega }_{10,1}^{\left(1\right)}& {\omega }_{82}^{\left(2\right)}-{\omega }_{10,2}^{\left(1\right)}& {\omega }_{83}^{\left(2\right)}-{\omega }_{10,3}^{\left(1\right)}& {\omega }_{64}^{\left(3\right)}& {\omega }_{85}^{\left(2\right)}-{\omega }_{10,5}^{\left(1\right)}& {\omega }_{87}^{\left(2\right)}-{\omega }_{10,9}^{\left(1\right)}\\ {\omega }_{91}^{\left(2\right)}-{\omega }_{11,1}^{\left(1\right)}& {\omega }_{92}^{\left(2\right)}-{\omega }_{11,2}^{\left(1\right)}& {\omega }_{93}^{\left(2\right)}-{\omega }_{11,3}^{\left(1\right)}& {\omega }_{74}^{\left(3\right)}& {\omega }_{95}^{\left(2\right)}-{\omega }_{11,5}^{\left(1\right)}& {\omega }_{97}^{\left(2\right)}-{\omega }_{11,9}^{\left(1\right)}\\ {\omega }_{81}^{\left(3\right)}& {\omega }_{82}^{\left(3\right)}& {\omega }_{83}^{\left(3\right)}& {\omega }_{84}^{\left(3\right)}& {\omega }_{10,5}^{\left(2\right)}-{\omega }_{12,5}^{\left(1\right)}& {\omega }_{85}^{\left(3\right)}\\ {\omega }_{11,1}^{\left(2\right)}-{\omega }_{13,1}^{\left(1\right)}& {\omega }_{11,2}^{\left(2\right)}-{\omega }_{13,2}^{\left(1\right)}& {\omega }_{11,3}^{\left(2\right)}-{\omega }_{13,3}^{\left(1\right)}& {\omega }_{11,4}^{\left(2\right)}-{\omega }_{13,4}^{\left(1\right)}& {\omega }_{11,5}^{\left(2\right)}-{\omega }_{13,5}^{\left(1\right)}& {\omega }_{11,7}^{\left(2\right)}-{\omega }_{13,9}^{\left(1\right)}\\ {\omega }_{91}^{\left(3\right)}-{\omega }_{51}^{\left(4\right)}+X_{31}^{\left(5\right)}& {\omega }_{92}^{\left(3\right)}-{\omega }_{52}^{\left(4\right)}& {\omega }_{93}^{\left(3\right)}-X_{73}^{\left(4\right)}& {\omega }_{94}^{\left(3\right)}& X_{11,5}^{\left(3\right)}& {\omega }_{95}^{\left(3\right)}-{\omega }_{53}^{\left(4\right)}+X_{32}^{\left(5\right)}\\ {\omega }_{10,1}^{\left(3\right)}-{\omega }_{61}^{\left(4\right)}& {\omega }_{10,2}^{\left(3\right)}-{\omega }_{62}^{\left(4\right)}& {\omega }_{10,3}^{\left(3\right)}-X_{83}^{\left(4\right)}& {\omega }_{10,4}^{\left(3\right)}& X_{12,5}^{\left(3\right)}& {\omega }_{10,5}^{\left(3\right)}-{\omega }_{63}^{\left(4\right)}\\ {\omega }_{11,1}^{\left(3\right)}-X_{91}^{\left(4\right)}& {\omega }_{11,2}^{\left(3\right)}-X_{92}^{\left(4\right)}& {\omega }_{11,3}^{\left(3\right)}-X_{93}^{\left(4\right)}& {\omega }_{11,4}^{\left(3\right)}& X_{13,5}^{\left(3\right)}& {\omega }_{11,5}^{\left(3\right)}-X_{94}^{\left(4\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Y_{21}^{\left(3\right)}=\left(\begin{array}{cccccc}{\omega }_{12,1}^{\left(3\right)}& {\omega }_{12,2}^{\left(3\right)}& {\omega }_{12,3}^{\left(3\right)}& {\omega }_{12,4}^{\left(3\right)}& X_{14,5}^{\left(3\right)}& {\omega }_{12,5}^{\left(3\right)}\\ X_{15,1}^{\left(3\right)}& X_{15,2}^{\left(3\right)}& X_{15,3}^{\left(3\right)}& X_{15,4}^{\left(3\right)}& X_{15,5}^{\left(3\right)}& X_{15,6}^{\left(3\right)}\\ {\omega }_{19,1}^{\left(2\right)}& {\omega }_{19,2}^{\left(2\right)}& {\omega }_{19,3}^{\left(2\right)}& {\omega }_{19,4}^{\left(2\right)}& {\omega }_{19,5}^{\left(2\right)}& {\omega }_{19,7}^{\left(2\right)}\\ {\omega }_{20,1}^{\left(2\right)}& {\omega }_{20,2}^{\left(2\right)}& {\omega }_{20,3}^{\left(2\right)}& {\omega }_{20,4}^{\left(2\right)}& {\omega }_{20,5}^{\left(2\right)}& {\omega }_{20,7}^{\left(2\right)}\\ {\omega }_{21,1}^{\left(2\right)}& {\omega }_{21,2}^{\left(2\right)}& {\omega }_{21,3}^{\left(2\right)}& {\omega }_{21,4}^{\left(2\right)}& {\omega }_{21,5}^{\left(2\right)}& {\omega }_{21,7}^{\left(2\right)}\\ {\omega }_{22,1}^{\left(2\right)}& {\omega }_{22,2}^{\left(2\right)}& {\omega }_{22,3}^{\left(2\right)}& {\omega }_{22,4}^{\left(2\right)}& {\omega }_{22,5}^{\left(2\right)}& {\omega }_{22,7}^{\left(2\right)}\\ {\omega }_{23,1}^{\left(2\right)}& {\omega }_{23,2}^{\left(2\right)}& {\omega }_{23,3}^{\left(2\right)}& {\omega }_{23,4}^{\left(2\right)}& {\omega }_{23,5}^{\left(2\right)}& {\omega }_{23,7}^{\left(2\right)}\\ {\omega }_{17,1}^{\left(3\right)}-{\omega }_{91}^{\left(4\right)}+X_{51}^{\left(5\right)}& {\omega }_{17,2}^{\left(3\right)}-{\omega }_{92}^{\left(4\right)}& {\omega }_{17,3}^{\left(3\right)}-X_{13,3}^{\left(4\right)}& {\omega }_{17,4}^{\left(3\right)}& X_{21,5}^{\left(3\right)}& {\omega }_{17,5}^{\left(3\right)}-{\omega }_{93}^{\left(4\right)}+X_{52}^{\left(5\right)}\\ {\omega }_{18,1}^{\left(3\right)}-{\omega }_{10,1}^{\left(4\right)}& {\omega }_{18,2}^{\left(3\right)}-{\omega }_{10,2}^{\left(4\right)}& {\omega }_{18,3}^{\left(3\right)}-X_{14,3}^{\left(4\right)}& {\omega }_{18,4}^{\left(3\right)}& X_{22,5}^{\left(3\right)}& {\omega }_{18,5}^{\left(3\right)}-{\omega }_{10,3}^{\left(4\right)}\\ {\omega }_{19,1}^{\left(3\right)}-X_{15,1}^{\left(4\right)}& {\omega }_{19,2}^{\left(3\right)}-X_{15,2}^{\left(4\right)}& {\omega }_{19,3}^{\left(3\right)}-X_{15,3}^{\left(4\right)}& {\omega }_{19,4}^{\left(3\right)}& X_{23,5}^{\left(3\right)}& {\omega }_{19,5}^{\left(3\right)}-X_{15,4}^{\left(4\right)}\\ {\omega }_{20,1}^{\left(3\right)}& {\omega }_{20,2}^{\left(3\right)}& {\omega }_{20,3}^{\left(3\right)}& {\omega }_{20,4}^{\left(3\right)}& X_{24,5}^{\left(3\right)}& {\omega }_{20,5}^{\left(3\right)}\\ X_{25,1}^{\left(3\right)}& X_{25,2}^{\left(3\right)}& X_{25,3}^{\left(3\right)}& X_{25,4}^{\left(3\right)}& X_{25,5}^{\left(3\right)}& X_{25,6}^{\left(3\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Y_{12}^{\left(3\right)}=\left(\begin{array}{cccccc}{\omega }_{18}^{\left(2\right)}-{\omega }_{1,10}^{\left(1\right)}& {\omega }_{19}^{\left(2\right)}-{\omega }_{1,11}^{\left(1\right)}& {\omega }_{18}^{\left(3\right)}& {\omega }_{1,11}^{\left(2\right)}-{\omega }_{1,13}^{\left(1\right)}& {\omega }_{19}^{\left(3\right)}-{\omega }_{15}^{\left(4\right)}+X_{13}^{\left(5\right)}& {\omega }_{1,10}^{\left(3\right)}-{\omega }_{16}^{\left(4\right)}\\ {\omega }_{28}^{\left(2\right)}-{\omega }_{2,10}^{\left(1\right)}& {\omega }_{29}^{\left(2\right)}-{\omega }_{2,11}^{\left(1\right)}& {\omega }_{28}^{\left(3\right)}& {\omega }_{2,11}^{\left(2\right)}-{\omega }_{2,13}^{\left(1\right)}& {\omega }_{29}^{\left(3\right)}-{\omega }_{25}^{\left(4\right)}& {\omega }_{2,10}^{\left(3\right)}-{\omega }_{26}^{\left(4\right)}\\ {\omega }_{38}^{\left(2\right)}-{\omega }_{3,10}^{\left(1\right)}& {\omega }_{39}^{\left(2\right)}-{\omega }_{3,11}^{\left(1\right)}& {\omega }_{38}^{\left(3\right)}& {\omega }_{3,11}^{\left(2\right)}-{\omega }_{3,13}^{\left(1\right)}& {\omega }_{39}^{\left(3\right)}-X_{37}^{\left(4\right)}& {\omega }_{3,10}^{\left(3\right)}-X_{38}^{\left(4\right)}\\ {\omega }_{46}^{\left(3\right)}& {\omega }_{47}^{\left(3\right)}& {\omega }_{48}^{\left(3\right)}& {\omega }_{4,11}^{\left(2\right)}-{\omega }_{4,13}^{\left(1\right)}& {\omega }_{49}^{\left(3\right)}& {\omega }_{4,10}^{\left(3\right)}\\ {\omega }_{58}^{\left(2\right)}-{\omega }_{5,10}^{\left(1\right)}& {\omega }_{59}^{\left(2\right)}-{\omega }_{5,11}^{\left(1\right)}& {\omega }_{5,10}^{\left(2\right)}-{\omega }_{5,12}^{\left(1\right)}& {\omega }_{5,11}^{\left(2\right)}-{\omega }_{5,13}^{\left(1\right)}& X_{5,11}^{\left(3\right)}& X_{5,12}^{\left(3\right)}\\ {\omega }_{78}^{\left(2\right)}-{\omega }_{9,10}^{\left(1\right)}& {\omega }_{79}^{\left(2\right)}-{\omega }_{9,11}^{\left(1\right)}& {\omega }_{58}^{\left(3\right)}& {\omega }_{7,11}^{\left(2\right)}-{\omega }_{9,13}^{\left(1\right)}& {\omega }_{59}^{\left(3\right)}-{\omega }_{35}^{\left(4\right)}+X_{23}^{\left(5\right)}& {\omega }_{5,10}^{\left(3\right)}-{\omega }_{36}^{\left(4\right)}\\ {\omega }_{88}^{\left(2\right)}-{\omega }_{10,10}^{\left(1\right)}& {\omega }_{89}^{\left(2\right)}-{\omega }_{10,11}^{\left(1\right)}& {\omega }_{68}^{\left(3\right)}& {\omega }_{8,11}^{\left(2\right)}-{\omega }_{10,13}^{\left(1\right)}& {\omega }_{69}^{\left(3\right)}-{\omega }_{45}^{\left(4\right)}& {\omega }_{6,10}^{\left(3\right)}-{\omega }_{46}^{\left(4\right)}\\ {\omega }_{98}^{\left(2\right)}-{\omega }_{11,10}^{\left(1\right)}& {\omega }_{99}^{\left(2\right)}-{\omega }_{11,11}^{\left(1\right)}& {\omega }_{78}^{\left(3\right)}& {\omega }_{9,11}^{\left(2\right)}-{\omega }_{11,13}^{\left(1\right)}& {\omega }_{79}^{\left(3\right)}-X_{67}^{\left(4\right)}& {\omega }_{7,10}^{\left(3\right)}-X_{68}^{\left(4\right)}\\ {\omega }_{86}^{\left(3\right)}& {\omega }_{87}^{\left(3\right)}& {\omega }_{88}^{\left(3\right)}& {\omega }_{10,11}^{\left(2\right)}-{\omega }_{12,13}^{\left(1\right)}& {\omega }_{89}^{\left(3\right)}& {\omega }_{8,10}^{\left(3\right)}\\ {\omega }_{11,8}^{\left(2\right)}-{\omega }_{13,10}^{\left(1\right)}& {\omega }_{11,9}^{\left(2\right)}-{\omega }_{13,11}^{\left(1\right)}& {\omega }_{11,10}^{\left(2\right)}-{\omega }_{13,12}^{\left(1\right)}& {\omega }_{11,11}^{\left(2\right)}-{\omega }_{13,13}^{\left(1\right)}& X_{10,11}^{\left(3\right)}& X_{10,12}^{\left(3\right)}\\ {\omega }_{96}^{\left(3\right)}-{\omega }_{54}^{\left(4\right)}& {\omega }_{97}^{\left(3\right)}-X_{76}^{\left(4\right)}& {\omega }_{98}^{\left(3\right)}& X_{11,10}^{\left(3\right)}& {\omega }_{99}^{\left(3\right)}-{\omega }_{55}^{\left(4\right)}+X_{33}^{\left(5\right)}& {\omega }_{9,10}^{\left(3\right)}-{\omega }_{56}^{\left(4\right)}\\ {\omega }_{10,6}^{\left(3\right)}-{\omega }_{64}^{\left(4\right)}& {\omega }_{10,7}^{\left(3\right)}-X_{86}^{\left(4\right)}& {\omega }_{10,8}^{\left(3\right)}& X_{12,10}^{\left(3\right)}& {\omega }_{10,9}^{\left(3\right)}-{\omega }_{65}^{\left(4\right)}& {\omega }_{10,10}^{\left(3\right)}-{\omega }_{66}^{\left(4\right)}\\ {\omega }_{11,6}^{\left(3\right)}-X_{95}^{\left(4\right)}& {\omega }_{11,7}^{\left(3\right)}-X_{96}^{\left(4\right)}& {\omega }_{11,8}^{\left(3\right)}& X_{13,10}^{\left(3\right)}& {\omega }_{11,9}^{\left(3\right)}-X_{97}^{\left(4\right)}& {\omega }_{11,10}^{\left(3\right)}-X_{98}^{\left(4\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Y_{22}^{\left(3\right)}=\left(\begin{array}{cccccc}{\omega }_{12,6}^{\left(3\right)}& {\omega }_{12,7}^{\left(3\right)}& {\omega }_{12,8}^{\left(3\right)}& X_{14,10}^{\left(3\right)}& {\omega }_{12,9}^{\left(3\right)}& {\omega }_{12,10}^{\left(3\right)}\\ X_{15,7}^{\left(3\right)}& X_{15,8}^{\left(3\right)}& X_{15,9}^{\left(3\right)}& X_{15,10}^{\left(3\right)}& X_{15,11}^{\left(3\right)}& X_{15,12}^{\left(3\right)}\\ {\omega }_{19,8}^{\left(2\right)}& {\omega }_{19,9}^{\left(2\right)}& {\omega }_{19,10}^{\left(2\right)}& {\omega }_{19,11}^{\left(2\right)}& {\omega }_{19,13}^{\left(2\right)}& {\omega }_{19,14}^{\left(2\right)}\\ {\omega }_{20,8}^{\left(2\right)}& {\omega }_{20,9}^{\left(2\right)}& {\omega }_{20,10}^{\left(2\right)}& {\omega }_{20,11}^{\left(2\right)}& {\omega }_{20,13}^{\left(2\right)}& {\omega }_{20,14}^{\left(2\right)}\\ {\omega }_{21,8}^{\left(2\right)}& {\omega }_{21,9}^{\left(2\right)}& {\omega }_{21,10}^{\left(2\right)}& {\omega }_{21,11}^{\left(2\right)}& {\omega }_{21,13}^{\left(2\right)}& {\omega }_{21,14}^{\left(2\right)}\\ {\omega }_{22,8}^{\left(2\right)}& {\omega }_{22,9}^{\left(2\right)}& {\omega }_{22,10}^{\left(2\right)}& {\omega }_{22,11}^{\left(2\right)}& {\omega }_{22,13}^{\left(2\right)}& {\omega }_{22,14}^{\left(2\right)}\\ {\omega }_{23,8}^{\left(2\right)}& {\omega }_{23,9}^{\left(2\right)}& {\omega }_{23,10}^{\left(2\right)}& {\omega }_{23,11}^{\left(2\right)}& {\omega }_{23,13}^{\left(2\right)}& {\omega }_{23,14}^{\left(2\right)}\\ {\omega }_{17,6}^{\left(3\right)}-{\omega }_{94}^{\left(4\right)}& {\omega }_{17,7}^{\left(3\right)}-X_{13,6}^{\left(4\right)}& {\omega }_{17,8}^{\left(3\right)}& X_{21,10}^{\left(3\right)}& {\omega }_{17,9}^{\left(3\right)}-{\omega }_{95}^{\left(4\right)}+X_{53}^{\left(5\right)}& {\omega }_{17,10}^{\left(3\right)}-{\omega }_{96}^{\left(4\right)}\\ {\omega }_{18,6}^{\left(3\right)}-{\omega }_{10,4}^{\left(4\right)}& {\omega }_{18,7}^{\left(3\right)}-X_{14,6}^{\left(4\right)}& {\omega }_{18,8}^{\left(3\right)}& X_{22,10}^{\left(3\right)}& {\omega }_{18,9}^{\left(3\right)}-{\omega }_{10,5}^{\left(4\right)}& {\omega }_{18,10}^{\left(3\right)}-{\omega }_{10,6}^{\left(4\right)}\\ {\omega }_{19,6}^{\left(3\right)}-X_{15,5}^{\left(4\right)}& {\omega }_{19,7}^{\left(3\right)}-X_{15,6}^{\left(4\right)}& {\omega }_{19,8}^{\left(3\right)}& X_{23,10}^{\left(3\right)}& {\omega }_{19,9}^{\left(3\right)}-X_{15,7}^{\left(4\right)}& {\omega }_{19,10}^{\left(3\right)}-X_{15,8}^{\left(4\right)}\\ {\omega }_{20,6}^{\left(3\right)}& {\omega }_{20,7}^{\left(3\right)}& {\omega }_{20,8}^{\left(3\right)}& X_{24,10}^{\left(3\right)}& {\omega }_{20,9}^{\left(3\right)}& {\omega }_{20,10}^{\left(3\right)}\\ X_{25,7}^{\left(3\right)}& X_{25,8}^{\left(3\right)}& X_{25,9}^{\left(3\right)}& X_{25,10}^{\left(3\right)}& X_{25,11}^{\left(3\right)}& X_{25,12}^{\left(3\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Y_{13}^{\left(3\right)}=\left(\begin{array}{cccccc}{\omega }_{1,11}^{\left(3\right)}-X_{19}^{\left(4\right)}& {\omega }_{1,12}^{\left(3\right)}& X_{1,15}^{\left(3\right)}& {\omega }_{1,19}^{\left(2\right)}& {\omega }_{1,20}^{\left(2\right)}& {\omega }_{1,21}^{\left(2\right)}\\ {\omega }_{2,11}^{\left(3\right)}-X_{29}^{\left(4\right)}& {\omega }_{2,12}^{\left(3\right)}& X_{2,15}^{\left(3\right)}& {\omega }_{2,19}^{\left(2\right)}& {\omega }_{2,20}^{\left(2\right)}& {\omega }_{2,21}^{\left(2\right)}\\ {\omega }_{3,11}^{\left(3\right)}-X_{39}^{\left(4\right)}& {\omega }_{3,12}^{\left(3\right)}& X_{3,15}^{\left(3\right)}& {\omega }_{3,19}^{\left(2\right)}& {\omega }_{3,20}^{\left(2\right)}& {\omega }_{3,21}^{\left(2\right)}\\ {\omega }_{4,11}^{\left(3\right)}& {\omega }_{4,12}^{\left(3\right)}& X_{4,15}^{\left(3\right)}& {\omega }_{4,19}^{\left(2\right)}& {\omega }_{4,20}^{\left(2\right)}& {\omega }_{4,21}^{\left(2\right)}\\ X_{5,13}^{\left(3\right)}& X_{5,14}^{\left(3\right)}& X_{5,15}^{\left(3\right)}& {\omega }_{5,19}^{\left(2\right)}& {\omega }_{5,20}^{\left(2\right)}& {\omega }_{5,21}^{\left(2\right)}\\ {\omega }_{5,11}^{\left(3\right)}-X_{49}^{\left(4\right)}& {\omega }_{5,12}^{\left(3\right)}& X_{6,15}^{\left(3\right)}& {\omega }_{7,19}^{\left(2\right)}& {\omega }_{7,20}^{\left(2\right)}& {\omega }_{7,21}^{\left(2\right)}\\ {\omega }_{6,11}^{\left(3\right)}-X_{59}^{\left(4\right)}& {\omega }_{6,12}^{\left(3\right)}& X_{7,15}^{\left(3\right)}& {\omega }_{8,19}^{\left(2\right)}& {\omega }_{8,20}^{\left(2\right)}& {\omega }_{8,21}^{\left(2\right)}\\ {\omega }_{7,11}^{\left(3\right)}-X_{69}^{\left(4\right)}& {\omega }_{7,12}^{\left(3\right)}& X_{8,15}^{\left(3\right)}& {\omega }_{9,19}^{\left(2\right)}& {\omega }_{9,20}^{\left(2\right)}& {\omega }_{9,21}^{\left(2\right)}\\ {\omega }_{8,11}^{\left(3\right)}& {\omega }_{8,12}^{\left(3\right)}& X_{9,15}^{\left(3\right)}& {\omega }_{10,19}^{\left(2\right)}& {\omega }_{10,20}^{\left(2\right)}& {\omega }_{10,21}^{\left(2\right)}\\ X_{10,13}^{\left(3\right)}& X_{10,14}^{\left(3\right)}& X_{10,15}^{\left(3\right)}& {\omega }_{11,19}^{\left(2\right)}& {\omega }_{11,20}^{\left(2\right)}& {\omega }_{11,21}^{\left(2\right)}\\ {\omega }_{9,11}^{\left(3\right)}-X_{79}^{\left(4\right)}& {\omega }_{9,12}^{\left(3\right)}& X_{11,15}^{\left(3\right)}& {\omega }_{13,19}^{\left(2\right)}& {\omega }_{13,20}^{\left(2\right)}& {\omega }_{13,21}^{\left(2\right)}\\ {\omega }_{10,11}^{\left(3\right)}-X_{89}^{\left(4\right)}& {\omega }_{10,12}^{\left(3\right)}& X_{12,15}^{\left(3\right)}& {\omega }_{14,19}^{\left(2\right)}& {\omega }_{14,20}^{\left(2\right)}& {\omega }_{14,21}^{\left(2\right)}\\ {\omega }_{11,11}^{\left(3\right)}-X_{99}^{\left(4\right)}& {\omega }_{11,12}^{\left(3\right)}& X_{13,15}^{\left(3\right)}& {\omega }_{15,19}^{\left(2\right)}& {\omega }_{15,20}^{\left(2\right)}& {\omega }_{15,21}^{\left(2\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Y_{23}^{\left(3\right)}=\left(\begin{array}{cccccc}{\omega }_{12,11}^{\left(3\right)}& {\omega }_{12,12}^{\left(3\right)}& X_{14,15}^{\left(3\right)}& {\omega }_{16,19}^{\left(2\right)}& {\omega }_{16,20}^{\left(2\right)}& {\omega }_{16,21}^{\left(2\right)}\\ X_{15,13}^{\left(3\right)}& X_{15,14}^{\left(3\right)}& X_{15,15}^{\left(3\right)}& {\omega }_{17,19}^{\left(2\right)}& {\omega }_{17,20}^{\left(2\right)}& {\omega }_{17,21}^{\left(2\right)}\\ {\omega }_{19,15}^{\left(2\right)}& {\omega }_{19,16}^{\left(2\right)}& {\omega }_{19,17}^{\left(2\right)}& {\omega }_{19,19}^{\left(2\right)}& {\omega }_{19,20}^{\left(2\right)}& {\omega }_{19,21}^{\left(2\right)}\\ {\omega }_{20,15}^{\left(2\right)}& {\omega }_{20,16}^{\left(2\right)}& {\omega }_{20,17}^{\left(2\right)}& {\omega }_{20,19}^{\left(2\right)}& {\omega }_{20,20}^{\left(2\right)}& {\omega }_{20,21}^{\left(2\right)}\\ {\omega }_{21,15}^{\left(2\right)}& {\omega }_{21,16}^{\left(2\right)}& {\omega }_{21,17}^{\left(2\right)}& {\omega }_{21,19}^{\left(2\right)}& {\omega }_{21,20}^{\left(2\right)}& {\omega }_{21,21}^{\left(2\right)}\\ {\omega }_{22,15}^{\left(2\right)}& {\omega }_{22,16}^{\left(2\right)}& {\omega }_{22,17}^{\left(2\right)}& {\omega }_{22,19}^{\left(2\right)}& {\omega }_{22,20}^{\left(2\right)}& {\omega }_{22,21}^{\left(2\right)}\\ {\omega }_{23,15}^{\left(2\right)}& {\omega }_{23,16}^{\left(2\right)}& {\omega }_{23,17}^{\left(2\right)}& {\omega }_{23,19}^{\left(2\right)}& {\omega }_{23,20}^{\left(2\right)}& {\omega }_{23,21}^{\left(2\right)}\\ {\omega }_{17,11}^{\left(3\right)}-X_{13,9}^{\left(4\right)}& {\omega }_{17,12}^{\left(3\right)}& X_{21,15}^{\left(3\right)}& {\omega }_{17,13}^{\left(3\right)}-{\omega }_{97}^{\left(4\right)}+X_{54}^{\left(5\right)}& {\omega }_{17,14}^{\left(3\right)}-{\omega }_{98}^{\left(4\right)}& {\omega }_{17,15}^{\left(3\right)}-X_{13,12}^{\left(4\right)}\\ {\omega }_{18,11}^{\left(3\right)}-X_{14,9}^{\left(4\right)}& {\omega }_{18,12}^{\left(3\right)}& X_{22,15}^{\left(3\right)}& {\omega }_{18,13}^{\left(3\right)}-{\omega }_{10,7}^{\left(4\right)}& {\omega }_{18,14}^{\left(3\right)}-{\omega }_{10,8}^{\left(4\right)}& {\omega }_{18,15}^{\left(3\right)}-X_{14,12}^{\left(4\right)}\\ {\omega }_{19,11}^{\left(3\right)}-X_{15,9}^{\left(4\right)}& {\omega }_{19,12}^{\left(3\right)}& X_{23,15}^{\left(3\right)}& {\omega }_{19,13}^{\left(3\right)}-X_{15,10}^{\left(4\right)}& {\omega }_{19,14}^{\left(3\right)}-X_{15,11}^{\left(4\right)}& {\omega }_{19,15}^{\left(3\right)}-X_{15,12}^{\left(4\right)}\\ {\omega }_{20,11}^{\left(3\right)}& {\omega }_{20,12}^{\left(3\right)}& X_{24,15}^{\left(3\right)}& {\omega }_{20,13}^{\left(3\right)}& {\omega }_{20,14}^{\left(3\right)}& {\omega }_{20,15}^{\left(3\right)}\\ X_{25,13}^{\left(3\right)}& X_{25,14}^{\left(3\right)}& X_{25,15}^{\left(3\right)}& X_{25,16}^{\left(3\right)}& X_{25,17}^{\left(3\right)}& X_{25,18}^{\left(3\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Y_{14}^{\left(3\right)}=\left(\begin{array}{ccccccc}{\omega }_{1,22}^{\left(2\right)}& {\omega }_{1,23}^{\left(2\right)}& {\omega }_{1,17}^{\left(3\right)}-{\omega }_{19}^{\left(4\right)}+X_{15}^{\left(5\right)}& {\omega }_{1,18}^{\left(3\right)}-{\omega }_{1,10}^{\left(4\right)}& {\omega }_{1,19}^{\left(3\right)}-X_{1,15}^{\left(4\right)}& {\omega }_{1,20}^{\left(3\right)}& X_{1,25}^{\left(3\right)}\\ {\omega }_{2,22}^{\left(2\right)}& {\omega }_{2,23}^{\left(2\right)}& {\omega }_{2,17}^{\left(3\right)}-{\omega }_{29}^{\left(4\right)}& {\omega }_{2,18}^{\left(3\right)}-{\omega }_{2,10}^{\left(4\right)}& {\omega }_{2,19}^{\left(3\right)}-X_{2,15}^{\left(4\right)}& {\omega }_{2,20}^{\left(3\right)}& X_{2,25}^{\left(3\right)}\\ {\omega }_{3,22}^{\left(2\right)}& {\omega }_{3,23}^{\left(2\right)}& {\omega }_{3,17}^{\left(3\right)}-X_{3,13}^{\left(4\right)}& {\omega }_{3,18}^{\left(3\right)}-X_{3,14}^{\left(4\right)}& {\omega }_{3,19}^{\left(3\right)}-X_{3,15}^{\left(4\right)}& {\omega }_{3,20}^{\left(3\right)}& X_{3,25}^{\left(3\right)}\\ {\omega }_{4,22}^{\left(2\right)}& {\omega }_{4,23}^{\left(2\right)}& {\omega }_{4,17}^{\left(3\right)}& {\omega }_{4,18}^{\left(3\right)}& {\omega }_{4,19}^{\left(3\right)}& {\omega }_{4,20}^{\left(3\right)}& X_{4,25}^{\left(3\right)}\\ {\omega }_{5,22}^{\left(2\right)}& {\omega }_{5,23}^{\left(2\right)}& X_{5,21}^{\left(3\right)}& X_{5,22}^{\left(3\right)}& X_{5,23}^{\left(3\right)}& X_{5,24}^{\left(3\right)}& X_{5,25}^{\left(3\right)}\\ {\omega }_{7,22}^{\left(2\right)}& {\omega }_{7,23}^{\left(2\right)}& {\omega }_{5,17}^{\left(3\right)}-{\omega }_{39}^{\left(4\right)}+X_{25}^{\left(5\right)}& {\omega }_{5,18}^{\left(3\right)}-{\omega }_{3,10}^{\left(4\right)}& {\omega }_{5,19}^{\left(3\right)}-X_{4,15}^{\left(4\right)}& {\omega }_{5,20}^{\left(3\right)}& X_{6,25}^{\left(3\right)}\\ {\omega }_{8,22}^{\left(2\right)}& {\omega }_{8,23}^{\left(2\right)}& {\omega }_{6,17}^{\left(3\right)}-{\omega }_{49}^{\left(4\right)}& {\omega }_{6,18}^{\left(3\right)}-{\omega }_{4,10}^{\left(4\right)}& {\omega }_{6,19}^{\left(3\right)}-X_{5,15}^{\left(4\right)}& {\omega }_{6,20}^{\left(3\right)}& X_{7,25}^{\left(3\right)}\\ {\omega }_{9,22}^{\left(2\right)}& {\omega }_{9,23}^{\left(2\right)}& {\omega }_{7,17}^{\left(3\right)}-X_{6,13}^{\left(4\right)}& {\omega }_{7,18}^{\left(3\right)}-X_{6,14}^{\left(4\right)}& {\omega }_{7,19}^{\left(3\right)}-X_{6,15}^{\left(4\right)}& {\omega }_{7,20}^{\left(3\right)}& X_{8,25}^{\left(3\right)}\\ {\omega }_{10,22}^{\left(2\right)}& {\omega }_{10,23}^{\left(2\right)}& {\omega }_{8,17}^{\left(3\right)}& {\omega }_{8,18}^{\left(3\right)}& {\omega }_{8,19}^{\left(3\right)}& {\omega }_{8,20}^{\left(3\right)}& X_{9,25}^{\left(3\right)}\\ {\omega }_{11,22}^{\left(2\right)}& {\omega }_{11,23}^{\left(2\right)}& X_{10,21}^{\left(3\right)}& X_{10,22}^{\left(3\right)}& X_{10,23}^{\left(3\right)}& X_{10,24}^{\left(3\right)}& X_{10,25}^{\left(3\right)}\\ {\omega }_{13,22}^{\left(2\right)}& {\omega }_{13,23}^{\left(2\right)}& {\omega }_{9,17}^{\left(3\right)}-{\omega }_{59}^{\left(4\right)}+X_{35}^{\left(5\right)}& {\omega }_{9,18}^{\left(3\right)}-{\omega }_{5,10}^{\left(4\right)}& {\omega }_{9,19}^{\left(3\right)}-X_{7,15}^{\left(4\right)}& {\omega }_{9,20}^{\left(3\right)}& X_{11,25}^{\left(3\right)}\\ {\omega }_{14,22}^{\left(2\right)}& {\omega }_{14,23}^{\left(2\right)}& {\omega }_{10,17}^{\left(3\right)}-{\omega }_{69}^{\left(4\right)}& {\omega }_{10,18}^{\left(3\right)}-{\omega }_{6,10}^{\left(4\right)}& {\omega }_{10,19}^{\left(3\right)}-X_{8,15}^{\left(4\right)}& {\omega }_{10,20}^{\left(3\right)}& X_{12,25}^{\left(3\right)}\\ {\omega }_{15,22}^{\left(2\right)}& {\omega }_{15,23}^{\left(2\right)}& {\omega }_{11,17}^{\left(3\right)}-X_{9,13}^{\left(4\right)}& {\omega }_{11,18}^{\left(3\right)}-X_{9,14}^{\left(4\right)}& {\omega }_{11,19}^{\left(3\right)}-X_{9,15}^{\left(4\right)}& {\omega }_{11,20}^{\left(3\right)}& X_{13,25}^{\left(3\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Y_{24}^{\left(3\right)}=\left(\begin{array}{ccccccc}{\omega }_{16,22}^{\left(2\right)}& {\omega }_{16,23}^{\left(2\right)}& {\omega }_{12,17}^{\left(3\right)}& {\omega }_{12,18}^{\left(3\right)}& {\omega }_{12,19}^{\left(3\right)}& {\omega }_{12,20}^{\left(3\right)}& X_{14,25}^{\left(3\right)}\\ {\omega }_{17,22}^{\left(2\right)}& {\omega }_{17,23}^{\left(2\right)}& X_{15,21}^{\left(3\right)}& X_{15,22}^{\left(3\right)}& X_{15,23}^{\left(3\right)}& X_{15,24}^{\left(3\right)}& X_{15,25}^{\left(3\right)}\\ {\omega }_{19,22}^{\left(2\right)}& {\omega }_{19,23}^{\left(2\right)}& {\omega }_{13,17}^{\left(3\right)}-{\omega }_{79}^{\left(4\right)}+X_{45}^{\left(5\right)}& {\omega }_{13,18}^{\left(3\right)}-{\omega }_{7,10}^{\left(4\right)}& {\omega }_{13,19}^{\left(3\right)}-X_{10,15}^{\left(4\right)}& {\omega }_{13,20}^{\left(3\right)}& X_{16,25}^{\left(3\right)}\\ {\omega }_{20,22}^{\left(2\right)}& {\omega }_{20,23}^{\left(2\right)}& {\omega }_{14,17}^{\left(3\right)}-{\omega }_{89}^{\left(4\right)}& {\omega }_{14,18}^{\left(3\right)}-{\omega }_{8,10}^{\left(4\right)}& {\omega }_{14,19}^{\left(3\right)}-X_{11,15}^{\left(4\right)}& {\omega }_{14,20}^{\left(3\right)}& X_{17,25}^{\left(3\right)}\\ {\omega }_{21,22}^{\left(2\right)}& {\omega }_{21,23}^{\left(2\right)}& {\omega }_{15,17}^{\left(3\right)}-X_{12,13}^{\left(4\right)}& {\omega }_{15,18}^{\left(3\right)}-X_{12,14}^{\left(4\right)}& {\omega }_{15,19}^{\left(3\right)}-X_{12,15}^{\left(4\right)}& {\omega }_{15,20}^{\left(3\right)}& X_{18,25}^{\left(3\right)}\\ {\omega }_{22,22}^{\left(2\right)}& {\omega }_{22,23}^{\left(2\right)}& {\omega }_{16,17}^{\left(3\right)}& {\omega }_{16,18}^{\left(3\right)}& {\omega }_{16,19}^{\left(3\right)}& {\omega }_{16,20}^{\left(3\right)}& X_{19,25}^{\left(3\right)}\\ {\omega }_{23,22}^{\left(2\right)}& {\omega }_{23,23}^{\left(2\right)}& X_{20,21}^{\left(3\right)}& X_{20,22}^{\left(3\right)}& X_{20,23}^{\left(3\right)}& X_{20,24}^{\left(3\right)}& X_{20,25}^{\left(3\right)}\\ {\omega }_{17,16}^{\left(3\right)}& X_{21,20}^{\left(3\right)}& {\omega }_{17,17}^{\left(3\right)}-{\omega }_{99}^{\left(4\right)}+X_{55}^{\left(5\right)}& {\omega }_{17,18}^{\left(3\right)}-{\omega }_{9,10}^{\left(4\right)}& {\omega }_{17,19}^{\left(3\right)}-X_{13,15}^{\left(4\right)}& {\omega }_{17,20}^{\left(3\right)}& X_{21,25}^{\left(3\right)}\\ {\omega }_{18,16}^{\left(3\right)}& X_{22,20}^{\left(3\right)}& {\omega }_{18,17}^{\left(3\right)}-{\omega }_{10,9}^{\left(4\right)}& {\omega }_{18,18}^{\left(3\right)}-{\omega }_{10,10}^{\left(4\right)}& {\omega }_{18,19}^{\left(3\right)}-X_{14,15}^{\left(4\right)}& {\omega }_{18,20}^{\left(3\right)}& X_{22,25}^{\left(3\right)}\\ {\omega }_{19,16}^{\left(3\right)}& X_{23,20}^{\left(3\right)}& {\omega }_{19,17}^{\left(3\right)}-X_{15,13}^{\left(4\right)}& {\omega }_{19,18}^{\left(3\right)}-X_{15,14}^{\left(4\right)}& {\omega }_{19,19}^{\left(3\right)}-X_{15,15}^{\left(4\right)}& {\omega }_{19,20}^{\left(3\right)}& X_{23,25}^{\left(3\right)}\\ {\omega }_{20,16}^{\left(3\right)}& X_{24,20}^{\left(3\right)}& {\omega }_{20,17}^{\left(3\right)}& {\omega }_{20,18}^{\left(3\right)}& {\omega }_{20,19}^{\left(3\right)}& {\omega }_{20,20}^{\left(3\right)}& X_{24,25}^{\left(3\right)}\\ X_{25,19}^{\left(3\right)}& X_{25,20}^{\left(3\right)}& X_{25,21}^{\left(3\right)}& X_{25,22}^{\left(3\right)}& X_{25,23}^{\left(3\right)}& X_{25,24}^{\left(3\right)}& X_{25,25}^{\left(3\right)}\end{array}\right).\end{array}$$

$$\begin{array}{c}\hfill \widehat{X_4}=\left(\begin{array}{ccc}{Y}_{11}^{\left(4\right)}& Y_{12}^{\left(4\right)}& Y_{13}^{\left(4\right)}\\ Y_{21}^{\left(4\right)}& Y_{22}^{\left(4\right)}& Y_{23}^{\left(4\right)}\end{array}\right),\end{array}$$

(70)

where

$$\begin{array}{cc}\hfill & Y_{11}^{\left(4\right)}=\hfill \\ \hfill & \left(\begin{array}{cccccc}{\omega }_{11}^{\left(4\right)}-X_{11}^{\left(5\right)}& {\omega }_{12}^{\left(4\right)}& X_{13}^{\left(4\right)}& {\omega }_{15}^{\left(3\right)}-{\omega }_{17}^{\left(2\right)}+{\omega }_{19}^{\left(1\right)}& {\omega }_{14}^{\left(4\right)}& {\omega }_{17}^{\left(3\right)}-{\omega }_{19}^{\left(2\right)}+{\omega }_{1,11}^{\left(1\right)}\\ {\omega }_{21}^{\left(4\right)}& {\omega }_{22}^{\left(4\right)}& X_{23}^{\left(4\right)}& {\omega }_{23}^{\left(4\right)}& {\omega }_{24}^{\left(4\right)}& {\omega }_{27}^{\left(3\right)}-{\omega }_{29}^{\left(2\right)}+{\omega }_{2,11}^{\left(1\right)}\\ X_{31}^{\left(4\right)}& X_{32}^{\left(4\right)}& X_{33}^{\left(4\right)}& {\omega }_{35}^{\left(3\right)}-{\omega }_{37}^{\left(2\right)}+{\omega }_{39}^{\left(1\right)}& {\omega }_{36}^{\left(3\right)}-{\omega }_{38}^{\left(2\right)}+{\omega }_{3,10}^{\left(1\right)}& {\omega }_{37}^{\left(3\right)}-{\omega }_{39}^{\left(2\right)}+{\omega }_{3,11}^{\left(1\right)}\\ {\omega }_{51}^{\left(3\right)}-{\omega }_{71}^{\left(2\right)}+{\omega }_{91}^{\left(1\right)}& {\omega }_{32}^{\left(4\right)}& {\omega }_{53}^{\left(3\right)}-{\omega }_{73}^{\left(2\right)}+{\omega }_{93}^{\left(1\right)}& {\omega }_{55}^{\left(3\right)}-{\omega }_{77}^{\left(2\right)}+{\omega }_{99}^{\left(1\right)}& {\omega }_{34}^{\left(4\right)}& {\omega }_{57}^{\left(3\right)}-{\omega }_{79}^{\left(2\right)}+{\omega }_{9,11}^{\left(1\right)}\\ {\omega }_{41}^{\left(4\right)}& {\omega }_{42}^{\left(4\right)}& {\omega }_{63}^{\left(3\right)}-{\omega }_{83}^{\left(2\right)}+{\omega }_{10,3}^{\left(1\right)}& {\omega }_{43}^{\left(4\right)}& {\omega }_{44}^{\left(4\right)}& {\omega }_{67}^{\left(3\right)}-{\omega }_{89}^{\left(2\right)}+{\omega }_{10,11}^{\left(1\right)}\\ {\omega }_{71}^{\left(3\right)}-{\omega }_{91}^{\left(2\right)}+{\omega }_{11,1}^{\left(1\right)}& {\omega }_{72}^{\left(3\right)}-{\omega }_{92}^{\left(2\right)}+{\omega }_{11,2}^{\left(1\right)}& {\omega }_{73}^{\left(3\right)}-{\omega }_{93}^{\left(2\right)}+{\omega }_{11,3}^{\left(1\right)}& {\omega }_{75}^{\left(3\right)}-{\omega }_{97}^{\left(2\right)}+{\omega }_{11,9}^{\left(1\right)}& {\omega }_{76}^{\left(3\right)}-{\omega }_{98}^{\left(2\right)}+{\omega }_{11,10}^{\left(1\right)}& {\omega }_{77}^{\left(3\right)}-{\omega }_{99}^{\left(2\right)}+{\omega }_{11,11}^{\left(1\right)}\\ {\omega }_{51}^{\left(4\right)}-X_{31}^{\left(5\right)}& {\omega }_{52}^{\left(4\right)}& X_{73}^{\left(4\right)}& {\omega }_{53}^{\left(4\right)}-X_{32}^{\left(5\right)}& {\omega }_{54}^{\left(4\right)}& X_{76}^{\left(4\right)}\\ {\omega }_{61}^{\left(4\right)}& {\omega }_{62}^{\left(4\right)}& X_{83}^{\left(4\right)}& {\omega }_{63}^{\left(4\right)}& {\omega }_{64}^{\left(4\right)}& X_{86}^{\left(4\right)}\\ X_{91}^{\left(4\right)}& X_{92}^{\left(4\right)}& X_{93}^{\left(4\right)}& X_{94}^{\left(4\right)}& X_{95}^{\left(4\right)}& X_{96}^{\left(4\right)}\\ {\omega }_{13,1}^{\left(3\right)}-{\omega }_{19,1}^{\left(2\right)}& {\omega }_{72}^{\left(4\right)}& {\omega }_{13,3}^{\left(3\right)}-{\omega }_{19,3}^{\left(2\right)}& {\omega }_{13,5}^{\left(3\right)}-{\omega }_{19,7}^{\left(2\right)}& {\omega }_{74}^{\left(4\right)}& {\omega }_{13,7}^{\left(3\right)}-{\omega }_{19,9}^{\left(2\right)}\\ {\omega }_{81}^{\left(4\right)}& {\omega }_{82}^{\left(4\right)}& {\omega }_{14,3}^{\left(3\right)}-{\omega }_{20,3}^{\left(2\right)}& {\omega }_{83}^{\left(4\right)}& {\omega }_{84}^{\left(4\right)}& {\omega }_{14,7}^{\left(3\right)}-{\omega }_{20,9}^{\left(2\right)}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill Y_{21}^{\left(4\right)}=\left(\begin{array}{cccccc}{\omega }_{15,1}^{\left(3\right)}-{\omega }_{21,1}^{\left(2\right)}& {\omega }_{15,2}^{\left(3\right)}-{\omega }_{21,2}^{\left(2\right)}& {\omega }_{15,3}^{\left(3\right)}-{\omega }_{21,3}^{\left(2\right)}& {\omega }_{15,5}^{\left(3\right)}-{\omega }_{21,7}^{\left(2\right)}& {\omega }_{15,6}^{\left(3\right)}-{\omega }_{21,8}^{\left(2\right)}& {\omega }_{15,7}^{\left(3\right)}-{\omega }_{21,9}^{\left(2\right)}\\ {\omega }_{91}^{\left(4\right)}-X_{51}^{\left(5\right)}& {\omega }_{92}^{\left(4\right)}& X_{13,3}^{\left(4\right)}& {\omega }_{93}^{\left(4\right)}-X_{52}^{\left(5\right)}& {\omega }_{94}^{\left(4\right)}& X_{13,6}^{\left(4\right)}\\ {\omega }_{10,1}^{\left(4\right)}& {\omega }_{10,2}^{\left(4\right)}& X_{14,3}^{\left(4\right)}& {\omega }_{10,3}^{\left(4\right)}& {\omega }_{10,4}^{\left(4\right)}& X_{14,6}^{\left(4\right)}\\ X_{15,1}^{\left(4\right)}& X_{15,2}^{\left(4\right)}& X_{15,3}^{\left(4\right)}& X_{15,4}^{\left(4\right)}& X_{15,5}^{\left(4\right)}& X_{15,6}^{\left(4\right)}\\ {\omega }_{21,1}^{\left(3\right)}& {\omega }_{11,2}^{\left(4\right)}& {\omega }_{21,3}^{\left(3\right)}& {\omega }_{21,5}^{\left(3\right)}& {\omega }_{11,4}^{\left(4\right)}& {\omega }_{21,7}^{\left(3\right)}\\ {\omega }_{12,1}^{\left(4\right)}& {\omega }_{12,2}^{\left(4\right)}& {\omega }_{22,3}^{\left(3\right)}& {\omega }_{12,3}^{\left(4\right)}& {\omega }_{12,4}^{\left(4\right)}& {\omega }_{22,7}^{\left(3\right)}\\ {\omega }_{23,1}^{\left(3\right)}& {\omega }_{23,2}^{\left(3\right)}& {\omega }_{23,3}^{\left(3\right)}& {\omega }_{23,5}^{\left(3\right)}& {\omega }_{23,6}^{\left(3\right)}& {\omega }_{23,7}^{\left(3\right)}\\ {\omega }_{13,1}^{\left(4\right)}-X_{71}^{\left(5\right)}& {\omega }_{13,2}^{\left(4\right)}& X_{19,3}^{\left(4\right)}& {\omega }_{13,3}^{\left(4\right)}-X_{72}^{\left(5\right)}& {\omega }_{13,4}^{\left(4\right)}& X_{19,6}^{\left(4\right)}\\ {\omega }_{14,1}^{\left(4\right)}& {\omega }_{14,2}^{\left(4\right)}& X_{20,3}^{\left(4\right)}& {\omega }_{14,3}^{\left(4\right)}& {\omega }_{14,4}^{\left(4\right)}& X_{20,6}^{\left(4\right)}\\ X_{21,1}^{\left(4\right)}& X_{21,2}^{\left(4\right)}& X_{21,3}^{\left(4\right)}& X_{21,4}^{\left(4\right)}& X_{21,5}^{\left(4\right)}& X_{21,6}^{\left(4\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Y_{12}^{\left(4\right)}=\left(\begin{array}{ccccccc}{\omega }_{15}^{\left(4\right)}-X_{13}^{\left(5\right)}& {\omega }_{16}^{\left(4\right)}& X_{19}^{\left(4\right)}& {\omega }_{1,13}^{\left(3\right)}-{\omega }_{1,19}^{\left(2\right)}& {\omega }_{18}^{\left(4\right)}& {\omega }_{1,15}^{\left(3\right)}-{\omega }_{1,21}^{\left(2\right)}& {\omega }_{19}^{\left(4\right)}-X_{15}^{\left(5\right)}\\ {\omega }_{25}^{\left(4\right)}& {\omega }_{26}^{\left(4\right)}& X_{29}^{\left(4\right)}& {\omega }_{27}^{\left(4\right)}& {\omega }_{28}^{\left(4\right)}& {\omega }_{2,15}^{\left(3\right)}-{\omega }_{2,21}^{\left(2\right)}& {\omega }_{29}^{\left(4\right)}\\ X_{37}^{\left(4\right)}& X_{38}^{\left(4\right)}& X_{39}^{\left(4\right)}& {\omega }_{3,13}^{\left(3\right)}-{\omega }_{3,19}^{\left(2\right)}& {\omega }_{3,14}^{\left(3\right)}-{\omega }_{3,20}^{\left(2\right)}& {\omega }_{3,15}^{\left(3\right)}-{\omega }_{3,21}^{\left(2\right)}& X_{3,13}^{\left(4\right)}\\ {\omega }_{35}^{\left(4\right)}-X_{23}^{\left(5\right)}& {\omega }_{36}^{\left(4\right)}& X_{49}^{\left(4\right)}& {\omega }_{5,13}^{\left(3\right)}-{\omega }_{7,19}^{\left(2\right)}& {\omega }_{38}^{\left(4\right)}& {\omega }_{5,15}^{\left(3\right)}-{\omega }_{7,21}^{\left(2\right)}& {\omega }_{39}^{\left(4\right)}-X_{25}^{\left(5\right)}\\ {\omega }_{45}^{\left(4\right)}& {\omega }_{46}^{\left(4\right)}& X_{59}^{\left(4\right)}& {\omega }_{47}^{\left(4\right)}& {\omega }_{48}^{\left(4\right)}& {\omega }_{6,15}^{\left(3\right)}-{\omega }_{8,21}^{\left(2\right)}& {\omega }_{49}^{\left(4\right)}\\ X_{67}^{\left(4\right)}& X_{68}^{\left(4\right)}& X_{69}^{\left(4\right)}& {\omega }_{7,13}^{\left(3\right)}-{\omega }_{9,19}^{\left(2\right)}& {\omega }_{7,14}^{\left(3\right)}-{\omega }_{9,20}^{\left(2\right)}& {\omega }_{7,15}^{\left(3\right)}-{\omega }_{9,21}^{\left(2\right)}& X_{6,13}^{\left(4\right)}\\ {\omega }_{55}^{\left(4\right)}-X_{33}^{\left(5\right)}& {\omega }_{56}^{\left(4\right)}& X_{79}^{\left(4\right)}& {\omega }_{9,13}^{\left(3\right)}-{\omega }_{13,19}^{\left(2\right)}& {\omega }_{58}^{\left(4\right)}& {\omega }_{9,15}^{\left(3\right)}-{\omega }_{13,21}^{\left(2\right)}& {\omega }_{59}^{\left(4\right)}-X_{35}^{\left(5\right)}\\ {\omega }_{65}^{\left(4\right)}& {\omega }_{66}^{\left(4\right)}& X_{89}^{\left(4\right)}& {\omega }_{67}^{\left(4\right)}& {\omega }_{68}^{\left(4\right)}& {\omega }_{10,15}^{\left(3\right)}-{\omega }_{14,21}^{\left(2\right)}& {\omega }_{69}^{\left(4\right)}\\ X_{97}^{\left(4\right)}& X_{98}^{\left(4\right)}& X_{99}^{\left(4\right)}& {\omega }_{11,13}^{\left(3\right)}-{\omega }_{15,19}^{\left(2\right)}& {\omega }_{11,14}^{\left(3\right)}-{\omega }_{15,20}^{\left(2\right)}& {\omega }_{11,15}^{\left(3\right)}-{\omega }_{15,21}^{\left(2\right)}& X_{9,13}^{\left(4\right)}\\ {\omega }_{13,9}^{\left(3\right)}-{\omega }_{19,13}^{\left(2\right)}& {\omega }_{76}^{\left(4\right)}& {\omega }_{13,11}^{\left(3\right)}-{\omega }_{19,15}^{\left(2\right)}& {\omega }_{13,13}^{\left(3\right)}-{\omega }_{19,19}^{\left(2\right)}& {\omega }_{78}^{\left(4\right)}& {\omega }_{13,15}^{\left(3\right)}-{\omega }_{19,21}^{\left(2\right)}& {\omega }_{79}^{\left(4\right)}-X_{45}^{\left(5\right)}\\ {\omega }_{85}^{\left(4\right)}& {\omega }_{86}^{\left(4\right)}& {\omega }_{14,11}^{\left(3\right)}-{\omega }_{20,15}^{\left(2\right)}& {\omega }_{87}^{\left(4\right)}& {\omega }_{88}^{\left(4\right)}& {\omega }_{14,15}^{\left(3\right)}-{\omega }_{20,21}^{\left(2\right)}& {\omega }_{89}^{\left(4\right)}\end{array}\right),\end{array}$$

$$\begin{array}{cc}\hfill & Y_{22}^{\left(4\right)}=\hfill \\ \hfill & \left(\begin{array}{ccccccc}{\omega }_{15,9}^{\left(3\right)}-{\omega }_{21,13}^{\left(2\right)}& {\omega }_{15,10}^{\left(3\right)}-{\omega }_{21,14}^{\left(2\right)}& {\omega }_{15,11}^{\left(3\right)}-{\omega }_{21,15}^{\left(2\right)}& {\omega }_{15,13}^{\left(3\right)}-{\omega }_{21,19}^{\left(2\right)}& {\omega }_{15,14}^{\left(3\right)}-{\omega }_{21,20}^{\left(2\right)}& {\omega }_{15,15}^{\left(3\right)}-{\omega }_{21,21}^{\left(2\right)}& X_{12,13}^{\left(4\right)}\\ {\omega }_{95}^{\left(4\right)}-X_{53}^{\left(5\right)}& {\omega }_{9,6}^{\left(4\right)}& X_{13,9}^{\left(4\right)}& {\omega }_{97}^{\left(4\right)}-X_{54}^{\left(5\right)}& {\omega }_{98}^{\left(4\right)}& X_{13,12}^{\left(4\right)}& {\omega }_{99}^{\left(4\right)}-X_{55}^{\left(5\right)}\\ {\omega }_{10,5}^{\left(4\right)}& {\omega }_{10,6}^{\left(4\right)}& X_{14,9}^{\left(4\right)}& {\omega }_{10,7}^{\left(4\right)}& {\omega }_{10,8}^{\left(4\right)}& X_{14,12}^{\left(4\right)}& {\omega }_{10,9}^{\left(4\right)}\\ X_{15,7}^{\left(4\right)}& X_{15,8}^{\left(4\right)}& X_{15,9}^{\left(4\right)}& X_{15,10}^{\left(4\right)}& X_{15,11}^{\left(4\right)}& X_{15,12}^{\left(4\right)}& X_{15,13}^{\left(4\right)}\\ {\omega }_{21,9}^{\left(3\right)}& {\omega }_{11,6}^{\left(4\right)}& {\omega }_{21,11}^{\left(3\right)}& {\omega }_{21,13}^{\left(3\right)}& {\omega }_{11,8}^{\left(4\right)}& {\omega }_{21,15}^{\left(3\right)}& {\omega }_{21,17}^{\left(3\right)}\\ {\omega }_{12,5}^{\left(4\right)}& {\omega }_{12,6}^{\left(4\right)}& {\omega }_{22,11}^{\left(3\right)}& {\omega }_{12,7}^{\left(4\right)}& {\omega }_{12,8}^{\left(4\right)}& {\omega }_{22,15}^{\left(3\right)}& {\omega }_{12,9}^{\left(4\right)}\\ {\omega }_{23,9}^{\left(3\right)}& {\omega }_{23,10}^{\left(3\right)}& {\omega }_{23,11}^{\left(3\right)}& {\omega }_{23,13}^{\left(3\right)}& {\omega }_{23,14}^{\left(3\right)}& {\omega }_{23,15}^{\left(3\right)}& {\omega }_{23,17}^{\left(3\right)}\\ {\omega }_{13,5}^{\left(4\right)}-X_{73}^{\left(5\right)}& {\omega }_{13,6}^{\left(4\right)}& X_{19,9}^{\left(4\right)}& {\omega }_{13,7}^{\left(4\right)}-X_{74}^{\left(5\right)}& {\omega }_{13,8}^{\left(4\right)}& X_{19,12}^{\left(4\right)}& {\omega }_{13,9}^{\left(4\right)}-X_{75}^{\left(5\right)}\\ {\omega }_{14,5}^{\left(4\right)}& {\omega }_{14,6}^{\left(4\right)}& X_{20,9}^{\left(4\right)}& {\omega }_{14,7}^{\left(4\right)}& {\omega }_{14,8}^{\left(4\right)}& X_{20,12}^{\left(4\right)}& {\omega }_{14,9}^{\left(4\right)}\\ X_{21,7}^{\left(4\right)}& X_{21,8}^{\left(4\right)}& X_{21,9}^{\left(4\right)}& X_{21,10}^{\left(4\right)}& X_{21,11}^{\left(4\right)}& X_{21,12}^{\left(4\right)}& X_{21,13}^{\left(4\right)}\end{array}\right),\hfill \end{array}$$

$$\begin{array}{c}\hfill Y_{13}^{\left(4\right)}=\left(\begin{array}{cccccccc}{\omega }_{1,10}^{\left(4\right)}& X_{1,15}^{\left(4\right)}& {\omega }_{1,21}^{\left(3\right)}& {\omega }_{1,12}^{\left(4\right)}& {\omega }_{1,23}^{\left(3\right)}& {\omega }_{1,13}^{\left(4\right)}-X_{17}^{\left(5\right)}& {\omega }_{1,14}^{\left(4\right)}& X_{1,21}^{\left(4\right)}\\ {\omega }_{2,10}^{\left(4\right)}& X_{2,15}^{\left(4\right)}& {\omega }_{2,11}^{\left(4\right)}& {\omega }_{2,12}^{\left(4\right)}& {\omega }_{2,23}^{\left(3\right)}& {\omega }_{2,13}^{\left(4\right)}& {\omega }_{2,14}^{\left(4\right)}& X_{2,21}^{\left(4\right)}\\ X_{3,14}^{\left(4\right)}& X_{3,15}^{\left(4\right)}& {\omega }_{3,21}^{\left(3\right)}& {\omega }_{3,22}^{\left(3\right)}& {\omega }_{3,23}^{\left(3\right)}& X_{3,19}^{\left(4\right)}& X_{3,20}^{\left(4\right)}& X_{3,21}^{\left(4\right)}\\ {\omega }_{3,10}^{\left(4\right)}& X_{4,15}^{\left(4\right)}& {\omega }_{5,21}^{\left(3\right)}& {\omega }_{3,12}^{\left(4\right)}& {\omega }_{5,23}^{\left(3\right)}& {\omega }_{3,13}^{\left(4\right)}-X_{27}^{\left(5\right)}& {\omega }_{3,14}^{\left(4\right)}& X_{4,21}^{\left(4\right)}\\ {\omega }_{4,10}^{\left(4\right)}& X_{5,15}^{\left(4\right)}& {\omega }_{4,11}^{\left(4\right)}& {\omega }_{4,12}^{\left(4\right)}& {\omega }_{6,23}^{\left(3\right)}& {\omega }_{4,13}^{\left(4\right)}& {\omega }_{4,14}^{\left(4\right)}& X_{5,21}^{\left(4\right)}\\ X_{6,14}^{\left(4\right)}& X_{6,15}^{\left(4\right)}& {\omega }_{7,21}^{\left(3\right)}& {\omega }_{7,22}^{\left(3\right)}& {\omega }_{7,23}^{\left(3\right)}& X_{6,19}^{\left(4\right)}& X_{6,20}^{\left(4\right)}& X_{6,21}^{\left(4\right)}\\ {\omega }_{5,10}^{\left(4\right)}& X_{7,15}^{\left(4\right)}& {\omega }_{9,21}^{\left(3\right)}& {\omega }_{5,12}^{\left(4\right)}& {\omega }_{9,23}^{\left(3\right)}& {\omega }_{5,13}^{\left(4\right)}-X_{37}^{\left(5\right)}& {\omega }_{5,14}^{\left(4\right)}& X_{7,21}^{\left(4\right)}\\ {\omega }_{6,10}^{\left(4\right)}& X_{8,15}^{\left(4\right)}& {\omega }_{6,11}^{\left(4\right)}& {\omega }_{6,12}^{\left(4\right)}& {\omega }_{10,23}^{\left(3\right)}& {\omega }_{6,13}^{\left(4\right)}& {\omega }_{6,14}^{\left(4\right)}& X_{8,21}^{\left(4\right)}\\ X_{9,14}^{\left(4\right)}& X_{9,15}^{\left(4\right)}& {\omega }_{11,21}^{\left(3\right)}& {\omega }_{11,22}^{\left(3\right)}& {\omega }_{11,23}^{\left(3\right)}& X_{9,19}^{\left(4\right)}& X_{9,20}^{\left(4\right)}& X_{9,21}^{\left(4\right)}\\ {\omega }_{7,10}^{\left(4\right)}& X_{10,15}^{\left(4\right)}& {\omega }_{13,21}^{\left(3\right)}& {\omega }_{7,12}^{\left(4\right)}& {\omega }_{13,23}^{\left(3\right)}& {\omega }_{7,13}^{\left(4\right)}-X_{47}^{\left(5\right)}& {\omega }_{7,14}^{\left(4\right)}& X_{10,21}^{\left(4\right)}\\ {\omega }_{8,10}^{\left(4\right)}& X_{11,15}^{\left(4\right)}& {\omega }_{8,11}^{\left(4\right)}& {\omega }_{8,12}^{\left(4\right)}& {\omega }_{14,23}^{\left(3\right)}& {\omega }_{8,13}^{\left(4\right)}& {\omega }_{8,14}^{\left(4\right)}& X_{11,21}^{\left(4\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Y_{23}^{\left(4\right)}=\left(\begin{array}{cccccccc}{X}_{12,14}^{\left(4\right)}& X_{12,15}^{\left(4\right)}& {\omega }_{15,21}^{\left(3\right)}& {\omega }_{15,22}^{\left(3\right)}& {\omega }_{15,23}^{\left(3\right)}& X_{12,19}^{\left(4\right)}& X_{12,20}^{\left(4\right)}& X_{12,21}^{\left(4\right)}\\ {\omega }_{9,10}^{\left(4\right)}& X_{13,15}^{\left(4\right)}& {\omega }_{17,21}^{\left(3\right)}& {\omega }_{9,12}^{\left(4\right)}& {\omega }_{17,23}^{\left(3\right)}& {\omega }_{9,13}^{\left(4\right)}-X_{57}^{\left(5\right)}& {\omega }_{9,14}^{\left(4\right)}& X_{13,21}^{\left(4\right)}\\ {\omega }_{10,10}^{\left(4\right)}& X_{14,15}^{\left(4\right)}& {\omega }_{10,11}^{\left(4\right)}& {\omega }_{10,12}^{\left(4\right)}& {\omega }_{18,23}^{\left(3\right)}& {\omega }_{10,13}^{\left(4\right)}& {\omega }_{10,14}^{\left(4\right)}& X_{14,21}^{\left(4\right)}\\ X_{15,14}^{\left(4\right)}& X_{15,15}^{\left(4\right)}& {\omega }_{19,21}^{\left(3\right)}& {\omega }_{19,22}^{\left(3\right)}& {\omega }_{19,23}^{\left(3\right)}& X_{15,19}^{\left(4\right)}& X_{15,20}^{\left(4\right)}& X_{15,21}^{\left(4\right)}\\ {\omega }_{11,10}^{\left(4\right)}& {\omega }_{21,19}^{\left(3\right)}& {\omega }_{21,21}^{\left(3\right)}& {\omega }_{11,12}^{\left(4\right)}& {\omega }_{21,23}^{\left(3\right)}& {\omega }_{11,13}^{\left(4\right)}-X_{67}^{\left(5\right)}& {\omega }_{11,14}^{\left(4\right)}& X_{16,21}^{\left(4\right)}\\ {\omega }_{12,10}^{\left(4\right)}& {\omega }_{22,19}^{\left(3\right)}& {\omega }_{12,11}^{\left(4\right)}& {\omega }_{12,12}^{\left(4\right)}& {\omega }_{22,23}^{\left(3\right)}& {\omega }_{12,13}^{\left(4\right)}& {\omega }_{12,14}^{\left(4\right)}& X_{17,21}^{\left(4\right)}\\ {\omega }_{23,18}^{\left(3\right)}& {\omega }_{23,19}^{\left(3\right)}& {\omega }_{23,21}^{\left(3\right)}& {\omega }_{23,22}^{\left(3\right)}& {\omega }_{23,23}^{\left(3\right)}& X_{18,19}^{\left(4\right)}& X_{18,20}^{\left(4\right)}& X_{18,21}^{\left(4\right)}\\ {\omega }_{13,10}^{\left(4\right)}& X_{19,15}^{\left(4\right)}& {\omega }_{13,11}^{\left(4\right)}-X_{76}^{\left(5\right)}& {\omega }_{13,12}^{\left(4\right)}& X_{19,18}^{\left(4\right)}& {\omega }_{13,13}^{\left(4\right)}-X_{77}^{\left(5\right)}& {\omega }_{13,14}^{\left(4\right)}& X_{19,21}^{\left(4\right)}\\ {\omega }_{14,10}^{\left(4\right)}& X_{20,15}^{\left(4\right)}& {\omega }_{14,11}^{\left(4\right)}& {\omega }_{14,12}^{\left(4\right)}& X_{20,18}^{\left(4\right)}& {\omega }_{14,13}^{\left(4\right)}& {\omega }_{14,14}^{\left(4\right)}& X_{20,21}^{\left(4\right)}\\ X_{21,14}^{\left(4\right)}& X_{21,15}^{\left(4\right)}& X_{21,16}^{\left(4\right)}& X_{21,17}^{\left(4\right)}& X_{21,18}^{\left(4\right)}& X_{21,19}^{\left(4\right)}& X_{21,20}^{\left(4\right)}& X_{21,21}^{\left(4\right)}\end{array}\right).\end{array}$$

$$\begin{array}{c}\hfill \widehat{X_5}=(Y_1^{\left(5\right)},Y_2^{\left(5\right)}),\end{array}$$

(71)

where

$$\begin{array}{c}\hfill Y_1^{\left(5\right)}=\left(\begin{array}{cccc}{X}_{11}^{\left(5\right)}& {\omega }_{13}^{\left(4\right)}-{\omega }_{15}^{\left(3\right)}+{\omega }_{17}^{\left(2\right)}-{\omega }_{19}^{\left(1\right)}& X_{13}^{\left(5\right)}& {\omega }_{17}^{\left(4\right)}-{\omega }_{1,13}^{\left(3\right)}+{\omega }_{1,19}^{\left(2\right)}\\ {\omega }_{31}^{\left(4\right)}-{\omega }_{51}^{\left(3\right)}+{\omega }_{71}^{\left(2\right)}-{\omega }_{91}^{\left(1\right)}& {\omega }_{33}^{\left(4\right)}-{\omega }_{55}^{\left(3\right)}+{\omega }_{77}^{\left(2\right)}-{\omega }_{99}^{\left(1\right)}& X_{23}^{\left(5\right)}& {\omega }_{37}^{\left(4\right)}-{\omega }_{5,13}^{\left(3\right)}+{\omega }_{7,19}^{\left(2\right)}\\ X_{31}^{\left(5\right)}& X_{32}^{\left(5\right)}& X_{33}^{\left(5\right)}& {\omega }_{57}^{\left(4\right)}-{\omega }_{9,13}^{\left(3\right)}+{\omega }_{13,19}^{\left(2\right)}\\ {\omega }_{71}^{\left(4\right)}-{\omega }_{13,1}^{\left(3\right)}+{\omega }_{19,1}^{\left(2\right)}& {\omega }_{73}^{\left(4\right)}-{\omega }_{13,5}^{\left(3\right)}+{\omega }_{19,7}^{\left(2\right)}& {\omega }_{75}^{\left(4\right)}-{\omega }_{13,9}^{\left(3\right)}+{\omega }_{19,13}^{\left(2\right)}& {\omega }_{77}^{\left(4\right)}-{\omega }_{13,13}^{\left(3\right)}+{\omega }_{19,19}^{\left(2\right)}\\ X_{51}^{\left(5\right)}& X_{52}^{\left(5\right)}& X_{53}^{\left(5\right)}& X_{54}^{\left(5\right)}\\ {\omega }_{11,1}^{\left(4\right)}-{\omega }_{21,1}^{\left(3\right)}& {\omega }_{11,3}^{\left(4\right)}-{\omega }_{21,5}^{\left(3\right)}& {\omega }_{11,5}^{\left(4\right)}-{\omega }_{21,9}^{\left(3\right)}& {\omega }_{11,7}^{\left(4\right)}-{\omega }_{21,13}^{\left(3\right)}\\ X_{71}^{\left(5\right)}& X_{72}^{\left(5\right)}& X_{73}^{\left(5\right)}& X_{74}^{\left(5\right)}\\ {\omega }_{15,1}^{\left(4\right)}& {\omega }_{15,3}^{\left(4\right)}& {\omega }_{15,5}^{\left(4\right)}& {\omega }_{15,7}^{\left(4\right)}\\ X_{91}^{\left(5\right)}& X_{92}^{\left(5\right)}& X_{93}^{\left(5\right)}& X_{94}^{\left(5\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Y_2^{\left(5\right)}=\left(\begin{array}{ccccc}{X}_{15}^{\left(5\right)}& {\omega }_{1,11}^{\left(4\right)}-{\omega }_{1,21}^{\left(3\right)}& X_{17}^{\left(5\right)}& {\omega }_{1,15}^{\left(4\right)}& X_{19}^{\left(5\right)}\\ X_{25}^{\left(5\right)}& {\omega }_{3,11}^{\left(4\right)}-{\omega }_{5,21}^{\left(3\right)}& X_{27}^{\left(5\right)}& {\omega }_{3,15}^{\left(4\right)}& X_{29}^{\left(5\right)}\\ X_{35}^{\left(5\right)}& {\omega }_{5,11}^{\left(4\right)}-{\omega }_{9,21}^{\left(3\right)}& X_{37}^{\left(5\right)}& {\omega }_{5,15}^{\left(4\right)}& X_{39}^{\left(5\right)}\\ X_{45}^{\left(5\right)}& {\omega }_{7,11}^{\left(4\right)}-{\omega }_{13,21}^{\left(3\right)}& X_{47}^{\left(5\right)}& {\omega }_{7,15}^{\left(4\right)}& X_{49}^{\left(5\right)}\\ X_{55}^{\left(5\right)}& {\omega }_{9,11}^{\left(4\right)}-{\omega }_{17,21}^{\left(3\right)}& X_{57}^{\left(5\right)}& {\omega }_{9,15}^{\left(4\right)}& X_{59}^{\left(5\right)}\\ {\omega }_{11,9}^{\left(4\right)}-{\omega }_{21,17}^{\left(3\right)}& {\omega }_{11,11}^{\left(4\right)}-{\omega }_{21,21}^{\left(3\right)}& X_{67}^{\left(5\right)}& {\omega }_{11,15}^{\left(4\right)}& X_{69}^{\left(5\right)}\\ X_{75}^{\left(5\right)}& X_{76}^{\left(5\right)}& X_{77}^{\left(5\right)}& {\omega }_{13,15}^{\left(4\right)}& X_{79}^{\left(5\right)}\\ {\omega }_{15,9}^{\left(4\right)}& {\omega }_{15,11}^{\left(4\right)}& {\omega }_{15,13}^{\left(4\right)}& {\omega }_{15,15}^{\left(4\right)}& X_{89}^{\left(5\right)}\\ X_{95}^{\left(5\right)}& X_{96}^{\left(5\right)}& X_{97}^{\left(5\right)}& X_{98}^{\left(5\right)}& X_{99}^{\left(5\right)}\end{array}\right).\end{array}$$

Algorithm 1: An algorithm to Find the General Solution to System (1)

1. Input $A_i\in {\mathbb{H}}^{p_i\times q_i}$, $B_i\in {\mathbb{H}}^{p_i\times q_{i+1}}$, $C_i\in {\mathbb{H}}^{t_i\times s_i}$, $D_i\in {\mathbb{H}}^{t_{i+1}\times s_i}$, ${\Omega }_i\in {\mathbb{H}}^{p_i\times s_i}$, $i=1,2,3,4$. 2. Compute the decompositions of (3) and (4) and derive the invertible quaternion matrices $P_i$, $S_i$, $Q_j$, $T_j$, $i=1,2,3,4$ and $j=1,2,3,4,5$. 3. Compute $\widehat{X_j}$ and $\widehat{{\Omega }_i}$ by $\widehat{X_j}=Q_j^{-1}{X}_j{T}_j^{-1}$ and $\widehat{{\Omega }_i}=P_i{\Omega }_i{S}_i$, $j=1,2,3,4,5$ and $i=1,2,3,4$. 4. Partition $\widehat{X_j}$, $j=1,2,3,4,5$ and $\widehat{{\Omega }_i}$, $i=1,2,3,4$. 5. Check whether (12)–(39) or (40)–(66) hold or not. If one of them holds, then proceed to the following steps. 6. Compute $\widehat{X_j}$ by (67)–(71), $j=1,2,3,4,5$. 7. Compute $X_j$ by $X_j=Q_j\widehat{X_j}{T}_j$, $j=1,2,3,4,5$.

4. A Numerical Example of System (1)

In this section, we show a numerical example of system (1).

 Example 1. 

Let

$$\begin{array}{c}\hfill A_1=\left(\begin{array}{ccc}1& -\mathbf{i}& \mathbf{k}\\ 2& -2\mathbf{i}& 2\mathbf{k}\\ \mathbf{j}& -\mathbf{k}& -\mathbf{i}\end{array}\right),A_2=\left(\begin{array}{ccc}1& -\mathbf{j}& 2+\mathbf{k}\\ \mathbf{j}& 0& 3+\mathbf{i}-\mathbf{k}\\ 2-\mathbf{k}& 3-\mathbf{i}+\mathbf{k}& 3\end{array}\right),A_3=\left(\begin{array}{ccc}1& \mathbf{j}& 2+2\mathbf{j}\\ -\mathbf{j}& 0& 3+\mathbf{i}\\ 2-\mathbf{j}& 3-\mathbf{i}& 3\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill A_4=\left(\begin{array}{ccc}-1& \mathbf{j}& \mathbf{i}-\mathbf{k}\\ \mathbf{k}& 2-\mathbf{k}& -\mathbf{i}\\ \mathbf{j}& \mathbf{j}+\mathbf{k}& -2+\mathbf{i}\end{array}\right),B_1=\left(\begin{array}{ccc}2& -\mathbf{i}& \mathbf{j}\\ 3& \mathbf{k}& 1\\ -2\mathbf{i}& 0& \mathbf{k}\end{array}\right),B_2=\left(\begin{array}{ccc}0& \mathbf{i}& \mathbf{j}\\ \mathbf{j}& 1& 2\mathbf{k}\\ -\mathbf{i}& 3& -1\end{array}\right)B_3=\left(\begin{array}{ccc}\mathbf{j}& \mathbf{k}& \mathbf{i}\\ \mathbf{k}& \mathbf{i}& \mathbf{j}\\ \mathbf{i}& \mathbf{j}& \mathbf{k}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill B_4=\left(\begin{array}{ccc}1& 2& -1\\ \mathbf{i}& 2\mathbf{i}& -\mathbf{i}\\ 1+\mathbf{i}& 2+2\mathbf{i}& -1-\mathbf{i}\end{array}\right),C_1=\left(\begin{array}{ccc}1& 2& \mathbf{j}\\ \mathbf{i}& 2\mathbf{i}& -\mathbf{k}\\ \mathbf{k}& 2\mathbf{k}& \mathbf{i}\end{array}\right),C_2=\left(\begin{array}{ccc}1& \mathbf{j}& 2-\mathbf{k}\\ -\mathbf{j}& 0& 3+\mathbf{k}\\ 2+\mathbf{k}& \mathbf{i}-\mathbf{k}& -3\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill C_3=\left(\begin{array}{ccc}1& \mathbf{j}& -1\\ \mathbf{j}& 0& \mathbf{j}\\ 2+2\mathbf{j}& 2\mathbf{j}& -2+2\mathbf{j}\end{array}\right),C_4=\left(\begin{array}{ccc}-1& \mathbf{k}& \mathbf{j}\\ \mathbf{j}& 2-\mathbf{k}& \mathbf{j}+\mathbf{k}\\ -\mathbf{i}-\mathbf{k}& \mathbf{i}& -2-\mathbf{i}\end{array}\right),D_1=\left(\begin{array}{ccc}2& 2\mathbf{i}& -2\mathbf{j}\\ -\mathbf{i}& -1& \mathbf{k}\\ \mathbf{j}& \mathbf{k}& 1\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill D_2=\left(\begin{array}{ccc}0& \mathbf{j}& \mathbf{i}\\ -\mathbf{i}& \mathbf{j}-\mathbf{k}& -1-\mathbf{i}+\mathbf{k}\\ 1+\mathbf{i}-\mathbf{k}& 3\mathbf{k}& \mathbf{k}\end{array}\right),D_3=\left(\begin{array}{ccc}-\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \mathbf{j}& 2+\mathbf{k}& 4-\mathbf{j}\\ \mathbf{k}& \mathbf{i}+2\mathbf{j}& \mathbf{j}\end{array}\right),D_4=\left(\begin{array}{ccc}1& -\mathbf{i}& \mathbf{k}\\ 2& 3-2\mathbf{i}& 2\mathbf{k}\\ -1& \mathbf{i}& -\mathbf{k}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\Omega }_1=\left(\begin{array}{ccc}-\mathbf{i}+7\mathbf{j}-\mathbf{k}& -12-3\mathbf{k}& 5-1\mathbf{i}+3\mathbf{k}\\ -7+6\mathbf{i}+14\mathbf{j}+4\mathbf{k}& -24-9\mathbf{i}+6\mathbf{j}-2\mathbf{k}& 10+4\mathbf{i}+7\mathbf{j}-2\mathbf{k}\\ -5\mathbf{i}+16\mathbf{j}-6\mathbf{k}& -8-2\mathbf{i}+13\mathbf{j}& 12-8\mathbf{i}-2\mathbf{j}+5\mathbf{k}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\Omega }_2=\left(\begin{array}{ccc}9+3\mathbf{i}+13\mathbf{j}+7\mathbf{k}& -7-\mathbf{i}+7\mathbf{j}& -18+13\mathbf{i}-9\mathbf{j}-15\mathbf{k}\\ 12+8\mathbf{i}+13\mathbf{j}+3\mathbf{k}& -2+3\mathbf{i}+8\mathbf{j}+3\mathbf{k}& -35+11\mathbf{i}-11\mathbf{j}-4\mathbf{k}\\ 10-4\mathbf{i}-2\mathbf{j}-17\mathbf{k}& -14+6\mathbf{i}+18\mathbf{j}+\mathbf{k}& 3+21\mathbf{i}+10\mathbf{j}+14\mathbf{k}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\Omega }_3=\left(\begin{array}{ccc}5+3\mathbf{i}+14\mathbf{j}+7\mathbf{k}& -20+4\mathbf{j}-7\mathbf{k}& -17+11\mathbf{i}-10\mathbf{j}-9\mathbf{k}\\ 12+10\mathbf{i}-3\mathbf{j}-2\mathbf{k}& -5-10\mathbf{i}+6\mathbf{j}-3\mathbf{k}& -12-6\mathbf{i}-7\mathbf{j}-2\mathbf{k}\\ 27-10\mathbf{i}+24\mathbf{j}-6\mathbf{k}& -5+16\mathbf{j}-9\mathbf{k}& -23+18\mathbf{i}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\Omega }_4=\left(\begin{array}{ccc}10+\mathbf{i}-\mathbf{j}+2\mathbf{k}& 18+2\mathbf{i}-9\mathbf{j}-10\mathbf{k}& 5-3\mathbf{j}+13\mathbf{k}\\ 1+3\mathbf{i}-\mathbf{j}-4\mathbf{k}& -6+9\mathbf{i}-4\mathbf{j}+9\mathbf{k}& 5-7\mathbf{i}-7\mathbf{j}-4\mathbf{k}\\ 7+8\mathbf{i}-\mathbf{j}-\mathbf{k}& 12+15\mathbf{i}-\mathbf{j}-15\mathbf{k}& 5-14\mathbf{j}+11\mathbf{k}\end{array}\right).\end{array}$$

Upon examination, (12)–(39) hold. Then, system (1) has a general solution. Note that

$$\begin{array}{c}\hfill X_1=\left(\begin{array}{ccc}1+\mathbf{j}& -2-\mathbf{i}+\mathbf{j}& \mathbf{i}+2\mathbf{k}\\ -2+\mathbf{i}& -1+2\mathbf{k}& \mathbf{i}-\mathbf{j}\\ \mathbf{j}+2\mathbf{k}& -\mathbf{i}-\mathbf{j}& 2\end{array}\right),X_2=\left(\begin{array}{ccc}\mathbf{j}+\mathbf{k}& \mathbf{i}& 2-\mathbf{j}\\ \mathbf{i}& 1-\mathbf{k}& -3-\mathbf{k}\\ 2\mathbf{i}-\mathbf{j}& -2-\mathbf{k}& 1+\mathbf{j}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill X_3=\left(\begin{array}{ccc}0& 2+\mathbf{i}& 3+\mathbf{k}\\ 2-\mathbf{i}& 1& -\mathbf{i}+\mathbf{j}\\ 3+\mathbf{k}& \mathbf{i}-\mathbf{j}& \mathbf{j}-\mathbf{k}\end{array}\right),X_4=\left(\begin{array}{ccc}2& -\mathbf{i}+\mathbf{j}& 1+\mathbf{i}\\ \mathbf{i}+\mathbf{j}& -2-\mathbf{j}& \mathbf{i}-\mathbf{k}\\ 1+\mathbf{j}& \mathbf{k}& 0\end{array}\right),X_5=\left(\begin{array}{ccc}-2-\mathbf{j}& 2+\mathbf{i}& \mathbf{j}-\mathbf{k}\\ 2-\mathbf{i}& 1-\mathbf{k}& 1-\mathbf{i}\\ \mathbf{k}& 1-\mathbf{k}& \mathbf{k}\end{array}\right),\end{array}$$

is a solution that satisfies system (1).

This experiment is conducted using MATLAB R2023B (MathWorks, Natick, MA, USA) running on a computer with the Windows 10 operating system.

5. Application of System (1) in Color Image Encryption and Decryption

In section, we make use of system (1) to develop a model which can be used to simultaneously encrypt four color images; this idea is similar to the idea put forward in [19].

The model simultaneously encrypting four color images is shown in Figure 1.

Let $X_1$, $X_2$, $X_3$ and $X_4$ stand for the four encrypted color images, and $X_5$ stand for a key used for encryption and decryption. It should be noted that $X_5$ can be a color image and can also be a general quaternion matrix with a proper size). The cipher consists of the invertible quaternion matrices $A_i$, $B_i$, $C_i$ and $D_i$, where $i=1,2,3,4$.

Then, we explain the encryption process. First, we randomly select an invertible $A_i$, $B_i$, $C_i$, $D_i(i=1,2,3,4)$ with a proper size, and select a key $X_5$. Then, we perform numerical calculations on $X_i(i=1,2,3,4,5)$ and $A_i$, $B_i$, $C_i$, $D_i(i=1,2,3,4)$ to obtain ${\Omega }_i(i=1,2,3,4)$ according to system (1). In this way, we can obtain the encrypted quaternion numerical matrices ${\Omega }_i(i=1,2,3,4)$. Based on this encryption process and the general solution to system (1), shown in Section 3. It is very difficult to correctly find the original color images when the keys are not disclosed. Hence, the encryption model is effective and secure, since there are an infinite number of choices of the free terms in the general solution.

In terms of the decryption process, we have utilized a picture to illustrate it. The model simultaneously decrypting four color images is shown in Figure 2.

In the decryption process, $DP{X}_i(i=1,2,3,4)$ represent the decrypted color images. Once the “Key” is given, we can reconstruct the original color images, starting with $DP{X}_4$ and going through to $DP{X}_1$ based on the process depicted in Figure 2.

Next, we give a numerical example. First, we select four color images to be encrypted and a key from the set of the sample pictures of MATLAB R2023B. The four images are “Indiancorn”, “Llama”, “Sevilla” and “Strawberries”, with the key “Yellowlily”. These original color images are shown in Figure 3.

Then we carry out the encryption process. Figure 4 shows that the encryption process makes the original image unrecognizable.

Next, we carry out the decryption process. The decrypted color images are shown in Figure 5.

It is easy to see from Figure 5 that the decrypted images are consistent with the original image in Figure 3. We use the Structural Similarity Index (SSIM) as an indicator to measure the decryption effect. Upon computation, the SSIM between the original images and the decrypted images is 1. This shows that the decryption process is effective and the decrypted images are of excellent quality.

Finally, it should be pointed out that encryption process based on the system of two-sided coupled generalized Sylvester quaternion matrix equations makes the decryption process more difficult without a key, and makes the decrypted color images have a stronger similarity with the original images.

This experiment is conducted using MATLAB R2023B (MathWorks, Natick, MA, USA) running on a computer with the Windows 10 operating system.

He et al. [19] made use of a system of Sylvester-type quaternion matrix equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{X}_1{A}_1-B_1{X}_2=C_1,\hfill \\ X_3{A}_2-B_2{X}_2=C_2,\hfill \\ X_3{A}_3-B_3{X}_4=C_3,\hfill \\ X_4{A}_4-B_4{X}_5=C_4,\hfill \\ X_6{A}_5-B_5{X}_5=C_5,\hfill \end{array}\right.\end{array}$$

(72)

to develop a frame to encrypt five color images simultaneously, where $A_i$, $B_i$, and $C_i$ are given quaternion matrices. The advantages of the encrypted frame developed by system (1) is more complex and safe than system (72). Beyond that, the decrypted images of the frame developed by system (1) is much more similar to the original images than the those of system (72), indicated by comparing the PSNR. The PSNR of the result of our frame is over 200, and is much larger than the result of the frame referred in [19]. The disadvantage of our frame is that it deals with four images simultaneously, which is less than the frame referred in [19]. If one wants to deal with more images, the simultaneous decomposition for more quaternion matrices should be considered.

6. Conclusions

In this paper, we study the solvability conditions and general solutions to a system of two-sided coupled Sylvester-type quaternion matrix equations

$$\begin{array}{c}\hfill A_i{X}_i{C}_i+B_i{X}_{i+1}{D}_i={\Omega }_i,i=1,2,3,4,\end{array}$$

in terms of the partition of quaternion matrices. Meanwhile, we show the equivalent relationship between the rank conditions of the coefficient matrices and the partitioned matrix conditions of the quaternion matrices. We also develop an algorithm to compute the general solution to the system. In addition, we give a numerical example. We also make use of system (1) to build a model that can be used to simultaneously encrypt and decrypt four color images.

We have shown that the simultaneous decomposition of multiple quaternion matrices play an important role in data storage and transmission. Our future work will include extending the simultaneous decomposition of eight quaternion matrices to the simultaneous decomposition of nine quaternion matrices or more, using the simultaneous decomposition of eight quaternion matrices to study the variations of system (1) to adapt to some specific physical systems.