A Color Image Encryption Model Based on a System of Quaternion Matrix Equations

Abstract

In the era of big data and multimedia communication, securing color images against unauthorized access and attacks is a pressing challenge. While quaternion-based models provide a unified representation for color images, most existing encryption schemes rely on single-image frameworks or lack the mathematical rigor to ensure both security and feasibility. To bridge this gap, this paper introduces a system of generalized Sylvester-type quaternion matrix equations as a novel encryption model. By using the equivalence canonical forms of five matrices arranged in a specific array, we provide necessary and sufficient conditions for the solvability of the generalized Sylvester-type quaternion matrix equation system, depending on the rank of the coefficient matrix. Numerical examples are provided to validate the obtained results. As an example of applications, we develop an encryption scheme for color images based on the proposed quaternion matrix equation system. Experimental results confirm the high feasibility of the proposed scheme. Notably, the proposed model supports dynamic key updates and multi-image secure transmission, making it highly adaptable for real-world applications. By integrating advanced quaternion matrix theory with practical image encryption, this work offers a scalable, secure, and mathematically sound approach to color image protection.

1. Introduction

Matrix decomposition constitutes a central technique in the theory of matrices. Among various decomposition methods, singular value decomposition(SVD) stands out as one of the most crucial matrix decompositions, with extensive applications in fields such as data analysis, image processing, and control theory. To address more complex problems, singular value decomposition has been extended to various generalized forms, such as the generalized singular value decomposition (GSVD), whose core idea is to extend the decomposition from a single matrix to multiple matrices.

Matrix decomposition over rings constitutes a significant topic in ring theory. However, due to the non-commutative nature of ring multiplication, research achievements in this area remain considerably limited compared to the well-established and rich theory of matrix decomposition over fields. Wang et al. [1] first investigated matrix triples with the same number of columns and established their equivalent canonical forms:

$$\begin{array}{cc}& m\\ \begin{array}{c}p\\ q\\ s\end{array}& \left(\begin{array}{c}C\\ D\\ E\end{array}\right)\end{array}$$

(1)

over an arbitrary division ring. Later in 2012, Wang et al. extended this research to the case of four matrices and proposed the corresponding equivalent canonical forms in [2]:

$$\begin{array}{cc}& m\hspace{1em}n\\ \begin{array}{c}p\\ q\\ s\end{array}& \left(\begin{array}{cc}B& C\\ & D\\ & E\end{array}\right)\end{array}$$

(2)

He et al. [3] further refined these canonical forms by specifying the exact dimensions of each constituent block. Subsequently, He et al. [4] provided an equivalent canonical form for the matrix array:

$$\begin{array}{cc}& m\hspace{1em}n\hspace{1em}p\\ \begin{array}{c}q\\ r\\ s\end{array}& \left(\begin{array}{ccc}A& B& \\ & C& \\ & D& E\end{array}\right)\end{array}$$

(3)

Sylvester-type matrix equations play an important role in control theory. Systematic results regarding their solvability conditions and general solution methods have been established (e.g., [5,6,7,8,9,10,11,12,13,14]). Based on the equivalent canonical form of five matrices, this paper establishes several solvability conditions for the following system of generalized Sylvester-type matrix equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{ccccc}AXF+& BYG& & =& \mathsf{\Phi },\\ & CYH& & =& \mathsf{\Psi },\\ & DYJ& +EZK& =& \mathsf{\Omega }.\end{array}\right.\end{array}$$

(4)

The algebra of quaternions constitutes an associative yet noncommutative division algebra. In recent years, quaternion theory and its matrix representations have gained significant importance in both theoretical and applied mathematics (e.g., [15,16,17,18,19,20,21,22]). In color image processing, a color image can be represented as a quaternion matrix [23]. In the era of big data, encrypting critical information is paramount, a requirement particularly pronounced for image data. This paper constructs an encryption scheme for color images utilizing the quaternion matrix system (4).

Recent advances in image encryption have leveraged diverse technologies, including chaotic systems, deep neural networks, quantum-inspired algorithms, optical encryption, and matrix-based algebraic approaches(e.g., [24,25,26,27,28,29,30]). Despite these developments, many schemes remain limited to single-image frameworks, lack dynamic key updating mechanisms, or suffer from high computational overhead. In contrast, our work integrates quaternion matrix theory with generalized Sylvester-type equations to enable simultaneous multi-image encryption with dynamic key support, offering a scalable and mathematically rigorous alternative. The main contributions of this paper are as follows:

1.

A solvability theory for a class of generalized Sylvester-type quaternion matrix equation systems is established. Based on the equivalent canonical form decomposition of a five-matrix array, we present for the first time a set of necessary and sufficient conditions for the solvability of system (4). These conditions are entirely characterized by the ranks of the given matrices, thereby providing a rigorous mathematical foundation for subsequent encryption applications.

2.

A novel color image encryption model based on a system of quaternion matrix equations is proposed. Utilizing the mathematical structure of system (4), this model simultaneously encodes two color images to be encrypted X and Y and an image serving as the key Z into a unified framework, achieving joint multi-image encryption and secure transmission.

3.

An encryption scheme supporting dynamic key updates is designed and implemented. The coefficient matrices $A,B,\dots ,K$, which act as a codebook in the model, can be dynamically generated or updated. This provides the system with dynamic security enhancement capabilities, overcoming the limitations of traditional single-image encryption schemes with static keys.

4.

The effectiveness and superiority of the proposed scheme are validated through numerical experiments and security analysis. Experimental results demonstrate that the proposed encryption scheme achieves near-ideal values for key security metrics, including the Number of Pixels Change Rate (NPCR), Unified Average Changing Intensity (UACI), information entropy, adjacent pixel correlation, and key sensitivity. This confirms its excellent encryption performance and robustness against common cryptographic attacks.

Let $\mathbb{R}$ and $\mathbb{H}$ denote the field of real numbers and the quaternion algebra, respectively.

$$\mathbb{H}=\{a_0+a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}\mid {\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=\mathbf{i}\mathbf{j}\mathbf{k}=-1, a_0,a_1,a_2,a_3\in \mathbb{R}\}.$$

We denote by ${\mathbb{H}}^{m\times n}$ and $G{L}_n\left(\mathbb{H}\right)$ the sets of all $m\times n$ matrices and all $n\times n$ invertible matrices over $\mathbb{H}$, respectively. Let $A\in {\mathbb{H}}^{m\times n}$, the symbols $I,\mathcal{R}\left(A\right),\mathcal{C}\left(A\right),\dim \mathcal{R}\left(A\right)$ represent the identity matrices of compatible dimensions, the left row space of A, the right column space of A, and the dimension of $\mathcal{R}\left(A\right)$, respectively. As is well known, $\dim \mathcal{C}\left(A\right)=\dim \mathcal{R}\left(A\right)$. This common value is referred to as the rank of the matrix A, denoted $r\left(A\right)$.

This article is organized as follows. In Section 2, we give the result of the simultaneous decomposition of the matrix array (3). In Section 3, by applying this decomposition, we establish the solvability conditions for the system of generalized Sylvester-type matrix Equation (4), which are determined by the ranks of the given matrices. In Section 4, some examples are provided to demonstrate the results. In Section 5, we investigate the application of the system of generalized Sylvester-type matrix Equation (4) to color image encryption.

2. The Equivalence Canonical Form of Matrix Array (3)

This section presents the equivalent canonical form of the matrix array

$$\begin{array}{c}\hfill \left(\begin{array}{ccc}A& B& \\ & C& \\ & D& E\end{array}\right).\end{array}$$

Lemma 1 

([4]). Consider a set of five matrices with compatible sizes: $A\in {\mathbb{H}}^{q\times m}$, $B\in {\mathbb{H}}^{q\times n}$, $C\in {\mathbb{H}}^{r\times n}$, $D\in {\mathbb{H}}^{s\times n}$ and $E\in {\mathbb{H}}^{s\times p}$. Then there exist $P_1\in G{L}_q\left(\mathbb{H}\right)$, $P_2\in G{L}_r\left(\mathbb{H}\right)$, $P_3\in G{L}_s\left(\mathbb{H}\right)$, $Q_1\in G{L}_m\left(\mathbb{H}\right)$, $Q_2\in G{L}_n\left(\mathbb{H}\right)$, and $Q_3\in G{L}_p\left(\mathbb{H}\right)$ such that

$$\begin{array}{c}\hfill P_{1}A{Q}_1=S_a, P_{1}B{Q}_2=S_b, P_{2}C{Q}_2=S_c, P_{3}D{Q}_2=S_d, P_{3}E{Q}_3=S_e, \end{array}$$

(5)

 where

(6)

 and

(7)

 with

$$\begin{array}{c}\hfill n_1=s_1+s_2+s_3+s_4+s_5+s_6,n_2=s_7+s_8+s_9+s_{10},\end{array}$$

$$\begin{array}{c}\hfill n_3=s_{11}+s_{12}+s_{13}+s_{14},n_4=s_8+s_{16}+s_{17}+s_{18},n_5=s_{19}+s_{20},\end{array}$$

$$\begin{array}{c}\hfill n_6=s_1+s_2+s_4+s_7+s_{21}+s_{22}+s_{23}+s_{24},n_7=s_{13}+s_{23}+s_{25}+s_{26},\end{array}$$

$$\begin{array}{c}\hfill n_8=s_1+s_4+s_5+s_{12}+s_{17}+s_{22}+s_{27}+s_{28}.\end{array}$$

We consider an equivalence canonical form of the matrix array in the form of the transpose of (3), that is,

$$\begin{array}{c}\hfill \left(\begin{array}{ccc}F& & \\ G& H& J\\ & & K\end{array}\right).\end{array}$$

We have the following lemma:

Lemma 2 

([4]). Given $F\in {\mathbb{H}}^{m_1\times q_1}$, $G\in {\mathbb{H}}^{n_1\times q_1}$, $H\in {\mathbb{H}}^{n_1\times r_1}$, $J\in {\mathbb{H}}^{n_1\times s_1}$ and $K\in {\mathbb{H}}^{p_1\times s_1}$. Then there exist $\widetilde{P_1}\in G{L}_{m_1}\left(\mathbb{H}\right)$, $\widetilde{P_2}\in G{L}_{n_1}\left(\mathbb{H}\right)$, $\widetilde{P_3}\in G{L}_{p_1}\left(\mathbb{H}\right)$, $\widetilde{Q_1}\in G{L}_{q_1}\left(\mathbb{H}\right)$, $\widetilde{Q_2}\in G{L}_{r_1}\left(\mathbb{H}\right)$, and $\widetilde{Q_3}\in G{L}_{s_1}\left(\mathbb{H}\right)$ such that

$$\begin{array}{c}\hfill \widetilde{P_1}F\widetilde{Q_1}=S_f, \widetilde{P_2}G\widetilde{Q_1}=S_g, \widetilde{P_2}H\widetilde{Q_2}=S_h, \widetilde{P_2}J\widetilde{Q_3}=S_j, \widetilde{P_3}K\widetilde{Q_3}=S_k, \end{array}$$

where $S_f,S_g,S_h,S_j$ and $S_k$ are block matrices in the form of the transpose of $S_a,S_b,S_c,S_d$ and $S_e$ in Lemma 1, respectively.

3. The Solvability Conditions of the System (4)

This section employs the equivalent canonical forms from Section 2 to establish the solvability conditions for the system (4).

Theorem 1. 

The system (4) is consistent if and only if the following rank equalities hold:

$$\begin{array}{c}\hfill r\left(\begin{array}{ccc}A& B& \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{cc}A& B\end{array}\right),r\left(\begin{array}{c}F\\ G\\ \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{c}F\\ G\end{array}\right),\end{array}$$

(8)

$$\begin{array}{c}\hfill r\left(\begin{array}{cc}C& \mathsf{\Psi }\end{array}\right)=r\left(C\right),r\left(\begin{array}{c}H\\ \mathsf{\Psi }\end{array}\right)=r\left(H\right),\end{array}$$

(9)

$$\begin{array}{c}\hfill r\left(\begin{array}{ccc}D& E& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cc}D& E\end{array}\right),r\left(\begin{array}{c}J\\ K\\ \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{c}J\\ K\end{array}\right),\end{array}$$

(10)

$$\begin{array}{c}\hfill r\left(\begin{array}{cc}0& G\\ A& \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{cc}0& G\\ A& 0\end{array}\right),r\left(\begin{array}{cc}0& F\\ B& \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{cc}0& F\\ B& 0\end{array}\right),\end{array}$$

(11)

$$\begin{array}{c}\hfill r\left(\begin{array}{cc}0& K\\ D& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cc}0& K\\ D& 0\end{array}\right),r\left(\begin{array}{cc}0& J\\ E& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cc}0& J\\ E& 0\end{array}\right),\end{array}$$

(12)

$$\begin{array}{c}\hfill r\left(\begin{array}{cccc}0& 0& G& H\\ A& B& -\mathsf{\Phi }& 0\\ 0& C& 0& \mathsf{\Psi }\end{array}\right)=r\left(\begin{array}{cc}G& H\end{array}\right)+r\left(\begin{array}{cc}A& B\\ 0& C\end{array}\right),r\left(\begin{array}{ccc}0& F& 0\\ 0& G& H\\ B& -\mathsf{\Phi }& 0\\ C& 0& \mathsf{\Psi }\end{array}\right)=r\left(\begin{array}{cc}F& 0\\ G& H\end{array}\right)+r\left(\begin{array}{c}B\\ C\end{array}\right),\end{array}$$

(13)

$$\begin{array}{c}\hfill r\left(\begin{array}{cccc}0& 0& H& J\\ C& 0& -\mathsf{\Psi }& 0\\ D& E& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cc}H& J\end{array}\right)+r\left(\begin{array}{cc}C& 0\\ D& E\end{array}\right),r\left(\begin{array}{ccc}0& H& J\\ 0& 0& K\\ C& -\mathsf{\Psi }& 0\\ D& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cc}H& J\\ 0& K\end{array}\right)+r\left(\begin{array}{c}C\\ D\end{array}\right),\end{array}$$

(14)

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{cccc}0& 0& G& J\\ 0& 0& 0& K\\ A& B& -\mathsf{\Phi }& 0\\ 0& D& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cc}G& J\\ 0& K\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{cc}A& B\\ 0& D\end{array}\end{array}\right),r\left(\begin{array}{cccc}0& 0& F& 0\\ 0& 0& G& J\\ B& 0& -\mathsf{\Phi }& 0\\ D& E& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cc}F& 0\\ G& J\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{cc}B& 0\\ D& E\end{array}\end{array}\right),\end{array}$$

(15)

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& 0& G& J\\ A& B& 0& -\mathsf{\Phi }& 0\\ 0& D& E& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cc}G& J\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{ccc}A& B& 0\\ 0& D& E\end{array}\end{array}\right),r\left(\begin{array}{c}\begin{array}{ccc}0& F& 0\\ 0& G& J\\ 0& 0& K\\ B& -\mathsf{\Phi }& 0\\ D& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cc}F& 0\\ G& J\\ 0& K\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{c}B\\ D\end{array}\end{array}\right),\end{array}$$

(16)

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{ccccccc}0& 0& 0& 0& G& H& J\\ A& B& 0& 0& \mathsf{\Phi }& 0& 0\\ 0& C& C& 0& 0& -\mathsf{\Psi }& 0\\ 0& 0& D& E& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccc}G& H& J\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{cccc}A& B& 0& 0\\ 0& C& C& 0\\ 0& 0& D& E\end{array}\end{array}\right),r\left(\begin{array}{c}\begin{array}{cccc}0& F& 0& 0\\ 0& G& H& 0\\ 0& 0& H& J\\ 0& 0& 0& K\\ B& \mathsf{\Phi }& 0& 0\\ C& 0& -\mathsf{\Psi }& 0\\ D& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccc}F& 0& 0\\ G& H& 0\\ 0& H& J\\ 0& 0& K\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{c}B\\ C\\ D\end{array}\end{array}\right),\end{array}$$

(17)

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{cccccc}0& 0& 0& G& H& J\\ 0& 0& 0& 0& 0& K\\ A& B& 0& \mathsf{\Phi }& 0& 0\\ 0& C& C& 0& -\mathsf{\Psi }& 0\\ 0& 0& D& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccc}G& H& J\\ 0& 0& K\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{ccc}A& B& 0\\ 0& C& C\\ 0& 0& D\end{array}\end{array}\right),r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& F& 0& 0\\ 0& 0& G& H& 0\\ 0& 0& 0& H& J\\ B& 0& \mathsf{\Phi }& 0& 0\\ C& 0& 0& -\mathsf{\Psi }& 0\\ D& E& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccc}F& 0& 0\\ G& H& 0\\ 0& H& J\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{cc}B& 0\\ C& 0\\ D& E\end{array}\end{array}\right),\end{array}$$

(18)

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{cccccc}0& 0& 0& F& 0& 0\\ 0& 0& 0& G& H& J\\ B& 0& 0& \mathsf{\Phi }& 0& 0\\ C& C& 0& 0& -\mathsf{\Psi }& 0\\ 0& D& E& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccc}F& 0& 0\\ G& H& J\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{ccc}B& 0& 0\\ C& C& 0\\ 0& D& E\end{array}\end{array}\right),r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& G& H& 0\\ 0& 0& 0& H& J\\ 0& 0& 0& 0& K\\ A& B& \mathsf{\Phi }& 0& 0\\ 0& C& 0& -\mathsf{\Psi }& 0\\ 0& D& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccc}G& H& 0\\ 0& H& J\\ 0& 0& K\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{cc}A& B\\ 0& C\\ 0& D\end{array}\end{array}\right),\end{array}$$

(19)

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{cccccc}0& 0& 0& G& H& 0\\ 0& 0& 0& 0& H& J\\ A& B& 0& \mathsf{\Phi }& 0& 0\\ 0& C& 0& 0& -\mathsf{\Psi }& 0\\ 0& D& E& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccc}G& H& 0\\ 0& H& J\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{ccc}A& B& 0\\ 0& C& 0\\ 0& D& E\end{array}\end{array}\right),r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& F& 0& 0\\ 0& 0& G& H& J\\ 0& 0& 0& 0& K\\ B& 0& \mathsf{\Phi }& 0& 0\\ C& C& 0& -\mathsf{\Psi }& 0\\ 0& D& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccc}F& 0& 0\\ G& H& J\\ 0& 0& K\end{array}\end{array}\right)+r\left(\begin{array}{c}\begin{array}{cc}B& 0\\ C& C\\ 0& D\end{array}\end{array}\right),\end{array}$$

(20)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cccccccc}0& 0& 0& 0& 0& F& 0& 0\\ 0& 0& 0& 0& G& 0& H& 0\\ 0& 0& 0& 0& 0& G& H& J\\ A& B& 0& 0& \mathsf{\Phi }& 0& 0& 0\\ 0& 0& B& 0& 0& -\mathsf{\Phi }& 0& 0\\ 0& C& 0& 0& 0& 0& -\mathsf{\Psi }& 0\\ 0& D& D& E& 0& 0& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cccc}0& F& 0& 0\\ G& 0& H& 0\\ 0& G& H& J\end{array}\right)+r\left(\begin{array}{cccc}A& B& 0& 0\\ 0& 0& B& 0\\ 0& C& 0& 0\\ 0& D& D& E\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccccccc}0& 0& 0& F& 0& 0& 0\\ 0& 0& 0& G& 0& H& J\\ 0& 0& 0& 0& G& 0& J\\ 0& 0& 0& 0& 0& 0& K\\ 0& B& 0& \mathsf{\Phi }& 0& 0& 0\\ A& 0& B& 0& -\mathsf{\Phi }& 0& 0\\ 0& C& C& 0& 0& -\mathsf{\Psi }& 0\\ 0& 0& D& 0& 0& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cccc}F& 0& 0& 0\\ G& 0& H& J\\ 0& G& 0& J\\ 0& 0& 0& K\end{array}\right)+r\left(\begin{array}{ccc}0& B& 0\\ A& 0& B\\ 0& C& C\\ 0& 0& D\end{array}\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cccccccc}0& 0& 0& 0& G& H& J& 0\\ 0& 0& 0& 0& 0& H& 0& J\\ 0& 0& 0& 0& 0& 0& K& 0\\ A& B& B& 0& \mathsf{\Phi }& 0& 0& 0\\ 0& C& 0& 0& 0& -\mathsf{\Psi }& 0& 0\\ 0& 0& D& 0& 0& 0& -\mathsf{\Omega }& 0\\ 0& D& 0& E& 0& 0& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cccc}G& H& J& 0\\ 0& H& 0& J\\ 0& 0& K& 0\end{array}\right)+r\left(\begin{array}{cccc}A& B& B& 0\\ 0& C& 0& 0\\ 0& 0& D& 0\\ 0& D& 0& E\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccccccc}0& 0& 0& F& 0& 0& 0\\ 0& 0& 0& G& H& 0& J\\ 0& 0& 0& G& 0& J& 0\\ 0& 0& 0& 0& 0& 0& K\\ B& 0& 0& \mathsf{\Phi }& 0& 0& 0\\ C& C& 0& 0& -\mathsf{\Psi }& 0& 0\\ D& 0& E& 0& 0& -\mathsf{\Omega }& 0\\ 0& D& 0& 0& 0& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cccc}F& 0& 0& 0\\ G& H& 0& J\\ G& 0& J& 0\\ 0& 0& 0& K\end{array}\right)+r\left(\begin{array}{ccc}B& 0& 0\\ C& C& 0\\ D& 0& E\\ 0& D& 0\end{array}\right),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & r\left(\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& F& 0& 0& 0\\ 0& 0& 0& 0& 0& G& 0& H& J& 0\\ 0& 0& 0& 0& 0& 0& G& H& 0& J\\ 0& 0& 0& 0& 0& 0& 0& 0& K& 0\\ A& B& 0& B& 0& \mathsf{\Phi }& 0& 0& 0& 0\\ 0& 0& B& 0& 0& 0& -\mathsf{\Phi }& 0& 0& 0\\ 0& C& 0& 0& 0& 0& 0& -\mathsf{\Psi }& 0& 0\\ 0& 0& 0& D& 0& 0& 0& 0& -\mathsf{\Omega }& 0\\ 0& D& D& 0& E& 0& 0& 0& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{ccccc}0& F& 0& 0& 0\\ G& 0& H& J& 0\\ 0& G& H& 0& J\\ 0& 0& 0& K& 0\end{array}\right)+r\left(\begin{array}{ccccc}A& B& 0& B& 0\\ 0& 0& B& 0& 0\\ 0& C& 0& 0& 0\\ 0& 0& 0& D& 0\\ 0& D& D& 0& E\end{array}\right),\hfill \\ \hfill & r\left(\begin{array}{ccccccccc}0& 0& 0& 0& F& 0& 0& 0& 0\\ 0& 0& 0& 0& G& 0& H& 0& J\\ 0& 0& 0& 0& 0& G& 0& 0& J\\ 0& 0& 0& 0& G& 0& 0& J& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& K\\ 0& B& 0& 0& \mathsf{\Phi }& 0& 0& 0& 0\\ A& 0& B& 0& 0& -\mathsf{\Phi }& 0& 0& 0\\ 0& C& C& 0& 0& 0& -\mathsf{\Psi }& 0& 0\\ 0& D& 0& E& 0& 0& 0& -\mathsf{\Omega }& 0\\ 0& 0& D& 0& 0& 0& 0& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{ccccc}F& 0& 0& 0& 0\\ G& 0& H& 0& J\\ 0& G& 0& 0& J\\ G& 0& 0& J& 0\\ 0& 0& 0& 0& K\end{array}\right)+r\left(\begin{array}{cccc}0& B& 0& 0\\ A& 0& B& 0\\ 0& C& C& 0\\ 0& D& 0& E\\ 0& 0& D& 0\end{array}\right).\hfill \end{array}$$

(23)

Proof. 

The “only if” part. Suppose $(X_0,Y_0,Z_0)$ is a solution to system (4). Thus, $(X_0,Y_0,Z_0)$ satisfies the system

$$\begin{array}{c}\hfill \left\{\begin{array}{ccccc}A{X}_{0}F+& B{Y}_{0}G& & =& \mathsf{\Phi },\\ & C{Y}_{0}H& & =& \mathsf{\Psi },\\ & D{Y}_{0}J& +E{Z}_{0}K& =& \mathsf{\Omega }.\end{array}\right.\end{array}$$

(24)

Using the matrix Equation (24) and elementary matrix operations, we demonstrate that the rank equalities (8)–(23) hold.

For the proof of the rank equalities (8)–(16), we note that

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccc}A& B& \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{ccc}A& B& A{X}_{0}F+B{Y}_{0}G\end{array}\right)\hfill \\ & =r\left[\left(\begin{array}{ccc}A& B& 0\end{array}\right)\left(\begin{array}{ccc}I& 0& X_{0}F\\ 0& I& Y_{0}G\\ 0& 0& I\end{array}\right)\right]\hfill \\ & =r\left(\begin{array}{cc}A& B\end{array}\right).\hfill \end{array}$$

The remaining rank equalities in (8)–(16) can be proved similarly.

For the proof of the rank equalities (17)–(23), we note that

$$\begin{array}{cc}\hfill & r\left(\begin{array}{ccccccc}0& 0& 0& 0& G& H& J\\ A& B& 0& 0& \mathsf{\Phi }& 0& 0\\ 0& C& C& 0& 0& -\mathsf{\Psi }& 0\\ 0& 0& D& E& 0& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{ccccccc}0& 0& 0& 0& G& H& J\\ A& B& 0& 0& A{X}_{0}F+B{Y}_{0}G& 0& 0\\ 0& C& C& 0& 0& -C{Y}_{0}H& 0\\ 0& 0& D& E& 0& 0& D{Y}_{0}J+E{Z}_{0}K\end{array}\right)\hfill \\ & =r\left[\left(\begin{array}{cccc}I& 0& 0& 0\\ B{Y}_0& I& 0& 0\\ 0& 0& I& 0\\ 0& 0& 0& I\end{array}\right)\left(\begin{array}{ccccccc}0& 0& 0& 0& G& H& J\\ A& B& 0& 0& 0& 0& 0\\ 0& C& C& 0& 0& 0& 0\\ 0& 0& D& E& 0& 0& 0\end{array}\right)\left(\begin{array}{ccccccc}I& 0& 0& 0& -X_{0}F& 0& 0\\ 0& I& 0& 0& 0& -Y_{0}H& 0\\ 0& 0& I& 0& 0& 0& Y_{0}J\\ 0& 0& 0& I& 0& 0& Z_{0}K\\ 0& 0& 0& 0& I& 0& 0\\ 0& 0& 0& 0& 0& I& 0\\ 0& 0& 0& 0& 0& 0& I\end{array}\right)\right]\hfill \\ & =r\left(\begin{array}{ccc}G& H& J\end{array}\right)+r\left(\begin{array}{cccc}A& B& 0& 0\\ 0& C& C& 0\\ 0& 0& D& E\end{array}\right).\hfill \end{array}$$

The remaining rank equalities in (17)–(23) can be proved similarly.

The “if” part. It follows from Lemmas 1 and 2 that the system of matrix Equation (4) is equivalent to the system of matrix equations

$$\begin{array}{c}\hfill \left\{\begin{array}{ccccc}{S}_a\left(Q_1^{-1}X{\widetilde{P_1}}^{-1}\right)S_f+& S_b\left(Q_2^{-1}Y{\widetilde{P_2}}^{-1}\right)S_g& & =& P_1\mathsf{\Phi }\widetilde{Q_1},\\ & S_c\left(Q_2^{-1}Y{\widetilde{P_2}}^{-1}\right)S_h& & =& P_2\mathsf{\Psi }\widetilde{Q_2},\\ & S_d\left(Q_2^{-1}Y{\widetilde{P_2}}^{-1}\right)S_j& +S_e\left(Q_3^{-1}Z{\widetilde{P_3}}^{-1}\right)S_k& =& P_3\mathsf{\Omega }\widetilde{Q_3}.\end{array}\right.\end{array}$$

(25)

Put

$$\begin{array}{c}\hfill P_1\mathsf{\Phi }\widetilde{Q_1}={\left({\mathsf{\Phi }}_{i,j}\right)}_{30\times 30},P_2\mathsf{\Psi }\widetilde{Q_2}={\left({\mathsf{\Psi }}_{i,j}\right)}_{22\times 22},P_3\mathsf{\Omega }\widetilde{Q_3}={\left({\mathsf{\Omega }}_{i,j}\right)}_{30\times 30},\end{array}$$

(26)

where the block rows of the matrix ${\left({\mathsf{\Phi }}_{i,j}\right)}_{30\times 30}$, ${\left({\mathsf{\Psi }}_{i,j}\right)}_{22\times 22}$ and ${\left({\mathsf{\Omega }}_{i,j}\right)}_{30\times 30}$ coincide with the block rows of $S_a$, $S_c$ and $S_d$, respectively, and their block columns coincide with the block columns of $S_f$, $S_h$, and $S_j$, respectively. Moreover, let

$$\begin{array}{c}\hfill \tilde{X}=Q_1^{-1}X{\widetilde{P_1}}^{-1}={\left(X_{i,j}\right)}_{16\times 16},\tilde{Y}=Q_2^{-1}Y{\widetilde{P_2}}^{-1}={\left(Y_{i,j}\right)}_{42\times 42},\tilde{Z}=Q_3^{-1}Z{\widetilde{P_3}}^{-1}={\left(Z_{i,j}\right)}_{30\times 30},\end{array}$$

(27)

where the block rows of the matrix ${\left(X_{i,j}\right)}_{16\times 16}$, ${\left(Y_{i,j}\right)}_{42\times 42}$ and ${\left(Z_{i,j}\right)}_{16\times 16}$ coincide with the block columns of $S_a$, $S_c$ and $S_d$, respectively, and their block columns coincide with the block rows of $S_f$, $S_h$ and $S_j$, respectively. Based on the form of $S_a,S_b,S_c,S_d,S_e,S_f,S_g,S_h,S_j$ and $S_k$, we can compute that (25) is equivalent to

$$\begin{array}{c}\hfill \left(\begin{array}{ccc}{\mathsf{\Phi }}_1& {\mathsf{\Phi }}_2& {\mathsf{\Phi }}_3\end{array}\right)={\left({\mathsf{\Phi }}_{i,j}\right)}_{30\times 30},\end{array}$$

(28)

$$\begin{array}{c}\hfill \left(\begin{array}{cccccc}{\mathsf{\Psi }}_1& {\mathsf{\Psi }}_2& {\mathsf{\Psi }}_3& {\mathsf{\Psi }}_4& {\mathsf{\Psi }}_5& {\mathsf{\Psi }}_6\end{array}\right)={\left({\mathsf{\Psi }}_{i,j}\right)}_{22\times 22},\end{array}$$

(29)

$$\begin{array}{c}\hfill \left(\begin{array}{ccc}{\mathsf{\Omega }}_1& {\mathsf{\Omega }}_2& {\mathsf{\Omega }}_3\end{array}\right)={\left({\mathsf{\Omega }}_{i,j}\right)}_{30\times 30}.\end{array}$$

(30)

The specific matrix is provided in the Appendix A.

Upon computations, we obtain that the matrix equations in (25) have a common solution if and only if

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\mathsf{\Phi }}_{1,30}\\ {\mathsf{\Phi }}_{2,30}\\ ⋮\\ {\mathsf{\Phi }}_{30,30}\end{array}\right)=0,\left(\begin{array}{cccc}{\mathsf{\Phi }}_{30,1}& {\mathsf{\Phi }}_{30,2}& \dots & {\mathsf{\Phi }}_{30,29}\end{array}\right)=0,\end{array}$$

(31)

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\mathsf{\Psi }}_{1,22}\\ {\mathsf{\Psi }}_{2,22}\\ ⋮\\ {\mathsf{\Psi }}_{22,22}\end{array}\right)=0,\left(\begin{array}{cccc}{\mathsf{\Psi }}_{22,1}& {\mathsf{\Psi }}_{22,2}& \dots & {\mathsf{\Psi }}_{22,21}\end{array}\right)=0,\end{array}$$

(32)

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\mathsf{\Omega }}_{1,30}\\ {\mathsf{\Omega }}_{2,30}\\ ⋮\\ {\mathsf{\Omega }}_{22,30}\end{array}\right)=0,\left(\begin{array}{cccc}{\mathsf{\Omega }}_{30,1}& {\mathsf{\Omega }}_{30,2}& \dots & {\mathsf{\Omega }}_{30,29}\end{array}\right)=0,\end{array}$$

(33)

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\mathsf{\Phi }}_{16,15}\\ {\mathsf{\Phi }}_{17,15}\\ ⋮\\ {\mathsf{\Phi }}_{30,15}\end{array}\right)=0,\left(\begin{array}{cccc}{\mathsf{\Phi }}_{15,16}& {\mathsf{\Phi }}_{15,17}& \dots & {\mathsf{\Phi }}_{15,30}\end{array}\right)=0,\end{array}$$

(34)

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\mathsf{\Omega }}_{5,29}\\ {\mathsf{\Omega }}_{6,29}\\ ⋮\\ {\mathsf{\Omega }}_{12,29}\end{array}\right)=0, \left(\begin{array}{c}{\mathsf{\Omega }}_{19,29}\\ {\mathsf{\Omega }}_{20,29}\\ {\mathsf{\Omega }}_{21,29}\\ {\mathsf{\Omega }}_{22,29}\end{array}\right)=0,\left(\begin{array}{c}{\mathsf{\Omega }}_{27,29}\\ {\mathsf{\Omega }}_{28,29}\end{array}\right)=0,\left\{\begin{array}{cc}\hfill & \left(\begin{array}{cccc}{\mathsf{\Omega }}_{29,5}& {\mathsf{\Omega }}_{29,6}& \dots & {\mathsf{\Omega }}_{29,12}\end{array}\right)=0,\hfill \\ \hfill & \left(\begin{array}{cccc}{\mathsf{\Omega }}_{29,19}& {\mathsf{\Omega }}_{29,20}& {\mathsf{\Omega }}_{29,21}& {\mathsf{\Omega }}_{29,22}\end{array}\right)=0,\hfill \\ \hfill & \left(\begin{array}{cc}{\mathsf{\Omega }}_{29,27}& {\mathsf{\Omega }}_{29,28}\end{array}\right)=0,\hfill \end{array}\right.\end{array}$$

(35)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \left(\begin{array}{cccccc}0& 0& 0& {\mathsf{\Psi }}_{12,19}& {\mathsf{\Psi }}_{12,20}& {\mathsf{\Psi }}_{12,21}\\ 0& 0& 0& {\mathsf{\Psi }}_{15,19}& {\mathsf{\Psi }}_{15,20}& {\mathsf{\Psi }}_{15,21}\\ 0& 0& 0& {\mathsf{\Psi }}_{18,19}& {\mathsf{\Psi }}_{18,20}& {\mathsf{\Psi }}_{18,21}\\ {\mathsf{\Psi }}_{19,12}& {\mathsf{\Psi }}_{19,15}& {\mathsf{\Psi }}_{19,18}& {\mathsf{\Psi }}_{19,19}& {\mathsf{\Psi }}_{19,20}& {\mathsf{\Psi }}_{19,21}\\ {\mathsf{\Psi }}_{20,12}& {\mathsf{\Psi }}_{20,15}& {\mathsf{\Psi }}_{20,18}& {\mathsf{\Psi }}_{20,19}& {\mathsf{\Psi }}_{20,20}& {\mathsf{\Psi }}_{20,21}\\ {\mathsf{\Psi }}_{21,12}& {\mathsf{\Psi }}_{21,15}& {\mathsf{\Psi }}_{21,18}& {\mathsf{\Psi }}_{21,19}& {\mathsf{\Psi }}_{21,20}& {\mathsf{\Psi }}_{21,21}\end{array}\right)=\left(\begin{array}{cccccc}0& 0& 0& {\mathsf{\Phi }}_{11,22}& {\mathsf{\Phi }}_{11,16}& {\mathsf{\Phi }}_{11,20}\\ 0& 0& 0& {\mathsf{\Phi }}_{3,22}& {\mathsf{\Phi }}_{3,16}& {\mathsf{\Phi }}_{3,20}\\ 0& 0& 0& {\mathsf{\Phi }}_{9,22}& {\mathsf{\Phi }}_{9,16}& {\mathsf{\Phi }}_{9,20}\\ {\mathsf{\Phi }}_{22,11}& {\mathsf{\Phi }}_{22,3}& {\mathsf{\Phi }}_{22,9}& {\mathsf{\Phi }}_{22,22}& {\mathsf{\Phi }}_{22,16}& {\mathsf{\Phi }}_{22,20}\\ {\mathsf{\Phi }}_{16,11}& {\mathsf{\Phi }}_{16,3}& {\mathsf{\Phi }}_{16,9}& {\mathsf{\Phi }}_{16,22}& {\mathsf{\Phi }}_{16,16}& {\mathsf{\Phi }}_{16,20}\\ {\mathsf{\Phi }}_{20,11}& {\mathsf{\Phi }}_{20,3}& {\mathsf{\Phi }}_{20,9}& {\mathsf{\Phi }}_{20,22}& {\mathsf{\Phi }}_{20,16}& {\mathsf{\Phi }}_{20,20}\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{cccc}{\mathsf{\Psi }}_{19,13}& {\mathsf{\Psi }}_{19,14}& {\mathsf{\Psi }}_{19,16}& {\mathsf{\Psi }}_{19,17}\\ {\mathsf{\Psi }}_{20,13}& {\mathsf{\Psi }}_{20,14}& {\mathsf{\Psi }}_{20,16}& {\mathsf{\Psi }}_{20,17}\end{array}\right)=\left(\begin{array}{cccc}{\mathsf{\Phi }}_{22,24}& {\mathsf{\Phi }}_{22,25}& {\mathsf{\Phi }}_{22,23}& {\mathsf{\Phi }}_{22,17}\\ {\mathsf{\Phi }}_{16,24}& {\mathsf{\Phi }}_{16,25}& {\mathsf{\Phi }}_{16,23}& {\mathsf{\Phi }}_{16,17}\end{array}\right)+\left(\begin{array}{cccc}{\mathsf{\Phi }}_{22,1}& {\mathsf{\Phi }}_{22,2}& {\mathsf{\Phi }}_{22,7}& {\mathsf{\Phi }}_{22,8}\\ {\mathsf{\Phi }}_{16,1}& {\mathsf{\Phi }}_{16,2}& {\mathsf{\Phi }}_{16,7}& {\mathsf{\Phi }}_{16,8}\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{cc}{\mathsf{\Psi }}_{13,19}& {\mathsf{\Psi }}_{13,20}\\ {\mathsf{\Psi }}_{14,19}& {\mathsf{\Psi }}_{14,20}\\ {\mathsf{\Psi }}_{16,19}& {\mathsf{\Psi }}_{16,20}\\ {\mathsf{\Psi }}_{17,19}& {\mathsf{\Psi }}_{17,20}\end{array}\right)=\left(\begin{array}{cc}{\mathsf{\Phi }}_{24,22}& {\mathsf{\Phi }}_{24,16}\\ {\mathsf{\Phi }}_{25,22}& {\mathsf{\Phi }}_{25,16}\\ {\mathsf{\Phi }}_{23,22}& {\mathsf{\Phi }}_{23,16}\\ {\mathsf{\Phi }}_{17,22}& {\mathsf{\Phi }}_{17,16}\end{array}\right)+\left(\begin{array}{cc}{\mathsf{\Phi }}_{1,22}& {\mathsf{\Phi }}_{1,16}\\ {\mathsf{\Phi }}_{2,22}& {\mathsf{\Phi }}_{2,16}\\ {\mathsf{\Phi }}_{7,22}& {\mathsf{\Phi }}_{7,16}\\ {\mathsf{\Phi }}_{8,22}& {\mathsf{\Phi }}_{8,16}\end{array}\right),\hfill \\ \hfill & {\mathsf{\Psi }}_{21,13}={\mathsf{\Phi }}_{20,24}+{\mathsf{\Phi }}_{20,1},\hfill \\ \hfill & {\mathsf{\Psi }}_{13,21}={\mathsf{\Phi }}_{24,20}+{\mathsf{\Phi }}_{1,20},\hfill \end{array}\right.\end{array}$$

(36)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \left(\begin{array}{cccccc}0& {\mathsf{\Psi }}_{4,11}& 0& {\mathsf{\Psi }}_{4,18}& 0& {\mathsf{\Psi }}_{4,21}\\ {\mathsf{\Psi }}_{11,4}& {\mathsf{\Psi }}_{11,11}& {\mathsf{\Psi }}_{11,15}& {\mathsf{\Psi }}_{11,18}& {\mathsf{\Psi }}_{11,20}& {\mathsf{\Psi }}_{11,21}\\ 0& {\mathsf{\Psi }}_{15,11}& 0& {\mathsf{\Psi }}_{15,18}& 0& {\mathsf{\Psi }}_{15,21}\\ {\mathsf{\Psi }}_{18,4}& {\mathsf{\Psi }}_{18,11}& {\mathsf{\Psi }}_{18,15}& {\mathsf{\Psi }}_{18,18}& {\mathsf{\Psi }}_{18,20}& {\mathsf{\Psi }}_{18,21}\\ 0& {\mathsf{\Psi }}_{20,11}& 0& {\mathsf{\Psi }}_{20,18}& 0& {\mathsf{\Psi }}_{20,21}\\ {\mathsf{\Psi }}_{21,4}& {\mathsf{\Psi }}_{21,11}& {\mathsf{\Psi }}_{21,15}& {\mathsf{\Psi }}_{21,18}& {\mathsf{\Psi }}_{21,20}& {\mathsf{\Psi }}_{21,21}\end{array}\right)=\left(\begin{array}{cccccc}0& {\mathsf{\Omega }}_{3,11}& 0& {\mathsf{\Omega }}_{3,21}& 0& {\mathsf{\Omega }}_{3,27}\\ {\mathsf{\Omega }}_{11,3}& {\mathsf{\Omega }}_{11,11}& {\mathsf{\Omega }}_{11,15}& {\mathsf{\Omega }}_{11,21}& {\mathsf{\Omega }}_{11,23}& {\mathsf{\Omega }}_{11,27}\\ 0& {\mathsf{\Omega }}_{15,11}& 0& {\mathsf{\Omega }}_{15,21}& 0& {\mathsf{\Omega }}_{15,27}\\ {\mathsf{\Omega }}_{21,3}& {\mathsf{\Omega }}_{21,11}& {\mathsf{\Omega }}_{21,15}& {\mathsf{\Omega }}_{21,21}& {\mathsf{\Omega }}_{21,23}& {\mathsf{\Omega }}_{21,27}\\ 0& {\mathsf{\Omega }}_{23,11}& 0& {\mathsf{\Omega }}_{23,21}& 0& {\mathsf{\Omega }}_{23,27}\\ {\mathsf{\Omega }}_{27,3}& {\mathsf{\Omega }}_{27,11}& {\mathsf{\Omega }}_{27,15}& {\mathsf{\Omega }}_{27,21}& {\mathsf{\Omega }}_{27,23}& {\mathsf{\Omega }}_{27,27}\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{ccc}{\mathsf{\Psi }}_{11,7}& {\mathsf{\Psi }}_{11,10}& {\mathsf{\Psi }}_{11,17}\\ {\mathsf{\Psi }}_{18,7}& {\mathsf{\Psi }}_{18,10}& {\mathsf{\Psi }}_{18,17}\\ {\mathsf{\Psi }}_{21,7}& {\mathsf{\Psi }}_{21,10}& {\mathsf{\Psi }}_{21,17}\end{array}\right)=\left(\begin{array}{ccc}{\mathsf{\Omega }}_{11,17}& {\mathsf{\Omega }}_{11,25}& {\mathsf{\Omega }}_{11,24}\\ {\mathsf{\Omega }}_{21,17}& {\mathsf{\Omega }}_{21,25}& {\mathsf{\Omega }}_{21,24}\\ {\mathsf{\Omega }}_{27,17}& {\mathsf{\Omega }}_{27,25}& {\mathsf{\Omega }}_{27,24}\end{array}\right)+\left(\begin{array}{ccc}{\mathsf{\Omega }}_{11,7}& {\mathsf{\Omega }}_{11,10}& {\mathsf{\Omega }}_{11,20}\\ {\mathsf{\Omega }}_{21,7}& {\mathsf{\Omega }}_{21,10}& {\mathsf{\Omega }}_{21,20}\\ {\mathsf{\Omega }}_{27,7}& {\mathsf{\Omega }}_{27,10}& {\mathsf{\Omega }}_{27,20}\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{ccc}{\mathsf{\Psi }}_{7,11}& {\mathsf{\Psi }}_{7,18}& {\mathsf{\Psi }}_{7,21}\\ {\mathsf{\Psi }}_{10,11}& {\mathsf{\Psi }}_{10,18}& {\mathsf{\Psi }}_{10,21}\\ {\mathsf{\Psi }}_{17,11}& {\mathsf{\Psi }}_{17,18}& {\mathsf{\Psi }}_{17,21}\end{array}\right)=\left(\begin{array}{ccc}{\mathsf{\Omega }}_{17,11}& {\mathsf{\Omega }}_{17,21}& {\mathsf{\Omega }}_{17,27}\\ {\mathsf{\Omega }}_{25,11}& {\mathsf{\Omega }}_{25,21}& {\mathsf{\Omega }}_{25,27}\\ {\mathsf{\Omega }}_{24,11}& {\mathsf{\Omega }}_{24,21}& {\mathsf{\Omega }}_{24,27}\end{array}\right)+\left(\begin{array}{ccc}{\mathsf{\Omega }}_{7,11}& {\mathsf{\Omega }}_{7,21}& {\mathsf{\Omega }}_{7,27}\\ {\mathsf{\Omega }}_{10,11}& {\mathsf{\Omega }}_{10,21}& {\mathsf{\Omega }}_{10,27}\\ {\mathsf{\Omega }}_{20,11}& {\mathsf{\Omega }}_{20,21}& {\mathsf{\Omega }}_{20,27}\end{array}\right),\hfill \end{array}\right.\end{array}$$

(37)

$$\begin{array}{c}\hfill \left(\begin{array}{c}\begin{array}{cccccccccc}0& 0& 0& 0& {\mathsf{\Phi }}_{7,16}& {\mathsf{\Phi }}_{7,17}& {\mathsf{\Phi }}_{7,18}& {\mathsf{\Phi }}_{7,19}& {\mathsf{\Phi }}_{7,20}& {\mathsf{\Phi }}_{7,21}\\ 0& 0& 0& 0& {\mathsf{\Phi }}_{8,16}& {\mathsf{\Phi }}_{8,17}& {\mathsf{\Phi }}_{8,18}& {\mathsf{\Phi }}_{8,19}& {\mathsf{\Phi }}_{8,20}& {\mathsf{\Phi }}_{8,21}\\ 0& 0& 0& 0& {\mathsf{\Phi }}_{9,16}& {\mathsf{\Phi }}_{9,17}& {\mathsf{\Phi }}_{9,18}& {\mathsf{\Phi }}_{9,19}& {\mathsf{\Phi }}_{9,20}& {\mathsf{\Phi }}_{9,21}\\ 0& 0& 0& 0& {\mathsf{\Phi }}_{10,16}& {\mathsf{\Phi }}_{10,17}& {\mathsf{\Phi }}_{10,18}& {\mathsf{\Phi }}_{10,19}& {\mathsf{\Phi }}_{10,20}& {\mathsf{\Phi }}_{10,21}\\ {\mathsf{\Phi }}_{16,7}& {\mathsf{\Phi }}_{16,8}& {\mathsf{\Phi }}_{16,9}& {\mathsf{\Phi }}_{16,10}& 0& 0& 0& 0& {\mathsf{\Phi }}_{16,20}& {\mathsf{\Phi }}_{16,21}\\ {\mathsf{\Phi }}_{17,7}& {\mathsf{\Phi }}_{17,8}& {\mathsf{\Phi }}_{17,9}& {\mathsf{\Phi }}_{17,10}& 0& 0& 0& 0& {\mathsf{\Phi }}_{17,20}& {\mathsf{\Phi }}_{17,21}\\ {\mathsf{\Phi }}_{18,7}& {\mathsf{\Phi }}_{18,8}& {\mathsf{\Phi }}_{18,9}& {\mathsf{\Phi }}_{18,10}& 0& 0& 0& 0& {\mathsf{\Phi }}_{18,20}& {\mathsf{\Phi }}_{18,21}\\ {\mathsf{\Phi }}_{19,7}& {\mathsf{\Phi }}_{19,8}& {\mathsf{\Phi }}_{19,9}& {\mathsf{\Phi }}_{19,10}& 0& 0& 0& 0& {\mathsf{\Phi }}_{19,20}& {\mathsf{\Phi }}_{19,21}\\ {\mathsf{\Phi }}_{20,7}& {\mathsf{\Phi }}_{20,8}& {\mathsf{\Phi }}_{20,9}& {\mathsf{\Phi }}_{20,10}& {\mathsf{\Phi }}_{20,16}& {\mathsf{\Phi }}_{20,17}& {\mathsf{\Phi }}_{20,18}& {\mathsf{\Phi }}_{20,19}& {\mathsf{\Phi }}_{20,20}& {\mathsf{\Phi }}_{20,21}\\ {\mathsf{\Phi }}_{21,7}& {\mathsf{\Phi }}_{21,8}& {\mathsf{\Phi }}_{21,9}& {\mathsf{\Phi }}_{21,10}& {\mathsf{\Phi }}_{21,16}& {\mathsf{\Phi }}_{21,17}& {\mathsf{\Phi }}_{21,18}& {\mathsf{\Phi }}_{21,19}& {\mathsf{\Phi }}_{21,20}& {\mathsf{\Phi }}_{21,21}\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{cccccccccc}0& 0& 0& 0& {\mathsf{\Omega }}_{19,23}& {\mathsf{\Omega }}_{19,24}& {\mathsf{\Omega }}_{19,25}& {\mathsf{\Omega }}_{19,26}& {\mathsf{\Omega }}_{19,27}& {\mathsf{\Omega }}_{19,28}\\ 0& 0& 0& 0& {\mathsf{\Omega }}_{20,23}& {\mathsf{\Omega }}_{20,24}& {\mathsf{\Omega }}_{20,25}& {\mathsf{\Omega }}_{20,26}& {\mathsf{\Omega }}_{20,27}& {\mathsf{\Omega }}_{20,28}\\ 0& 0& 0& 0& {\mathsf{\Omega }}_{21,23}& {\mathsf{\Omega }}_{21,24}& {\mathsf{\Omega }}_{21,25}& {\mathsf{\Omega }}_{21,26}& {\mathsf{\Omega }}_{21,27}& {\mathsf{\Omega }}_{21,28}\\ 0& 0& 0& 0& {\mathsf{\Omega }}_{22,23}& {\mathsf{\Omega }}_{22,24}& {\mathsf{\Omega }}_{22,25}& {\mathsf{\Omega }}_{22,26}& {\mathsf{\Omega }}_{22,27}& {\mathsf{\Omega }}_{22,28}\\ {\mathsf{\Omega }}_{23,19}& {\mathsf{\Omega }}_{23,20}& {\mathsf{\Omega }}_{23,21}& {\mathsf{\Omega }}_{23,22}& 0& 0& 0& 0& {\mathsf{\Omega }}_{23,27}& {\mathsf{\Omega }}_{23,28}\\ {\mathsf{\Omega }}_{24,19}& {\mathsf{\Omega }}_{24,20}& {\mathsf{\Omega }}_{24,21}& {\mathsf{\Omega }}_{24,22}& 0& 0& 0& 0& {\mathsf{\Omega }}_{24,27}& {\mathsf{\Omega }}_{24,28}\\ {\mathsf{\Omega }}_{25,19}& {\mathsf{\Omega }}_{25,20}& {\mathsf{\Omega }}_{25,21}& {\mathsf{\Omega }}_{25,22}& 0& 0& 0& 0& {\mathsf{\Omega }}_{25,27}& {\mathsf{\Omega }}_{25,28}\\ {\mathsf{\Omega }}_{26,19}& {\mathsf{\Omega }}_{26,20}& {\mathsf{\Omega }}_{26,21}& {\mathsf{\Omega }}_{26,22}& 0& 0& 0& 0& {\mathsf{\Omega }}_{26,27}& {\mathsf{\Omega }}_{26,28}\\ {\mathsf{\Omega }}_{27,19}& {\mathsf{\Omega }}_{27,20}& {\mathsf{\Omega }}_{27,21}& {\mathsf{\Omega }}_{27,22}& {\mathsf{\Omega }}_{27,23}& {\mathsf{\Omega }}_{27,24}& {\mathsf{\Omega }}_{27,25}& {\mathsf{\Omega }}_{27,26}& {\mathsf{\Omega }}_{27,27}& {\mathsf{\Omega }}_{27,28}\\ {\mathsf{\Omega }}_{28,19}& {\mathsf{\Omega }}_{28,20}& {\mathsf{\Omega }}_{28,21}& {\mathsf{\Omega }}_{28,22}& {\mathsf{\Omega }}_{28,23}& {\mathsf{\Omega }}_{28,24}& {\mathsf{\Omega }}_{28,25}& {\mathsf{\Omega }}_{28,26}& {\mathsf{\Omega }}_{28,27}& {\mathsf{\Omega }}_{28,28}\end{array}\end{array}\right),\end{array}$$

(38)

$$\begin{array}{c}\hfill \left(\begin{array}{c}\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& {\mathsf{\Phi }}_{1,20}& {\mathsf{\Phi }}_{1,21}\\ 0& 0& 0& 0& 0& 0& {\mathsf{\Phi }}_{2,20}& {\mathsf{\Phi }}_{2,21}\\ 0& 0& 0& 0& 0& 0& {\mathsf{\Phi }}_{3,20}& {\mathsf{\Phi }}_{3,21}\\ 0& 0& 0& 0& 0& 0& {\mathsf{\Phi }}_{4,20}& {\mathsf{\Phi }}_{4,21}\\ 0& 0& 0& 0& 0& 0& {\mathsf{\Phi }}_{5,20}& {\mathsf{\Phi }}_{5,21}\\ 0& 0& 0& 0& 0& 0& {\mathsf{\Phi }}_{6,20}& {\mathsf{\Phi }}_{6,21}\\ {\mathsf{\Phi }}_{20,1}& {\mathsf{\Phi }}_{20,2}& {\mathsf{\Phi }}_{20,3}& {\mathsf{\Phi }}_{20,4}& {\mathsf{\Phi }}_{20,5}& {\mathsf{\Phi }}_{20,6}& 0& 0\\ {\mathsf{\Phi }}_{21,1}& {\mathsf{\Phi }}_{21,2}& {\mathsf{\Phi }}_{21,3}& {\mathsf{\Phi }}_{21,4}& {\mathsf{\Phi }}_{21,5}& {\mathsf{\Phi }}_{21,6}& 0& 0\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& {\mathsf{\Omega }}_{13,27}& {\mathsf{\Omega }}_{13,28}\\ 0& 0& 0& 0& 0& 0& {\mathsf{\Omega }}_{14,27}& {\mathsf{\Omega }}_{14,28}\\ 0& 0& 0& 0& 0& 0& {\mathsf{\Omega }}_{15,27}& {\mathsf{\Omega }}_{15,28}\\ 0& 0& 0& 0& 0& 0& {\mathsf{\Omega }}_{16,27}& {\mathsf{\Omega }}_{16,28}\\ 0& 0& 0& 0& 0& 0& {\mathsf{\Omega }}_{17,27}& {\mathsf{\Omega }}_{17,28}\\ 0& 0& 0& 0& 0& 0& {\mathsf{\Omega }}_{18,27}& {\mathsf{\Omega }}_{18,28}\\ {\mathsf{\Omega }}_{27,13}& {\mathsf{\Omega }}_{27,14}& {\mathsf{\Omega }}_{27,15}& {\mathsf{\Omega }}_{27,16}& {\mathsf{\Omega }}_{27,17}& {\mathsf{\Omega }}_{27,18}& 0& 0\\ {\mathsf{\Omega }}_{28,13}& {\mathsf{\Omega }}_{28,14}& {\mathsf{\Omega }}_{28,15}& {\mathsf{\Omega }}_{28,16}& {\mathsf{\Omega }}_{28,17}& {\mathsf{\Omega }}_{28,18}& 0& 0\end{array}\end{array}\right),\end{array}$$

(39)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Psi }}_{21,2}\\ {\mathsf{\Psi }}_{21,3}\\ {\mathsf{\Psi }}_{21,5}\\ {\mathsf{\Psi }}_{21,8}\\ {\mathsf{\Psi }}_{21,14}\\ {\mathsf{\Psi }}_{21,16}\end{array}\end{array}\right)}^{\mathsf{T}}={\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Phi }}_{20,13}\\ {\mathsf{\Phi }}_{20,28}\\ {\mathsf{\Phi }}_{20,12}\\ {\mathsf{\Phi }}_{20,26}\\ {\mathsf{\Phi }}_{20,25}\\ {\mathsf{\Phi }}_{20,23}\end{array}\end{array}\right)}^{\mathsf{T}}+{\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Omega }}_{27,1}\\ {\mathsf{\Omega }}_{27,2}\\ {\mathsf{\Omega }}_{27,5}\\ {\mathsf{\Omega }}_{27,8}\\ {\mathsf{\Omega }}_{27,14}\\ {\mathsf{\Omega }}_{27,19}\end{array}\end{array}\right)}^{\mathsf{T}},\hfill \\ \hfill & \left(\begin{array}{c}\begin{array}{c}{\mathsf{\Psi }}_{2,21}\\ {\mathsf{\Psi }}_{3,21}\\ {\mathsf{\Psi }}_{5,21}\\ {\mathsf{\Psi }}_{8,21}\\ {\mathsf{\Psi }}_{14,21}\\ {\mathsf{\Psi }}_{16,21}\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Phi }}_{13,20}\\ {\mathsf{\Phi }}_{28,20}\\ {\mathsf{\Phi }}_{12,20}\\ {\mathsf{\Phi }}_{26,20}\\ {\mathsf{\Phi }}_{25,20}\\ {\mathsf{\Phi }}_{23,20}\end{array}\end{array}\right)+\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Omega }}_{1,27}\\ {\mathsf{\Omega }}_{2,27}\\ {\mathsf{\Omega }}_{5,27}\\ {\mathsf{\Omega }}_{8,27}\\ {\mathsf{\Omega }}_{14,27}\\ {\mathsf{\Omega }}_{19,27}\end{array}\end{array}\right),\hfill \\ \hfill & {\mathsf{\Psi }}_{21,6}={\mathsf{\Phi }}_{20,27}+{\mathsf{\Omega }}_{27,16}+{\mathsf{\Omega }}_{27,6},\hfill \\ \hfill & {\mathsf{\Psi }}_{6,21}={\mathsf{\Phi }}_{27,20}+{\mathsf{\Omega }}_{16,27}+{\mathsf{\Omega }}_{6,27},\hfill \end{array}\right.\end{array}$$

(40)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Psi }}_{20,5}\\ {\mathsf{\Psi }}_{20,7}\\ {\mathsf{\Psi }}_{20,8}\\ {\mathsf{\Psi }}_{20,10}\end{array}\end{array}\right)}^{\mathsf{T}}={\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Phi }}_{16,12}\\ {\mathsf{\Phi }}_{16,5}\\ {\mathsf{\Phi }}_{16,26}\\ {\mathsf{\Phi }}_{16,18}\end{array}\end{array}\right)}^{\mathsf{T}}+{\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Omega }}_{23,5}\\ {\mathsf{\Omega }}_{23,7}\\ {\mathsf{\Omega }}_{23,8}\\ {\mathsf{\Omega }}_{23,10}\end{array}\end{array}\right)}^{\mathsf{T}},\hfill \\ \hfill & \left(\begin{array}{c}\begin{array}{c}{\mathsf{\Psi }}_{5,20}\\ {\mathsf{\Psi }}_{7,20}\\ {\mathsf{\Psi }}_{8,20}\\ {\mathsf{\Psi }}_{10,20}\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Phi }}_{12,16}\\ {\mathsf{\Phi }}_{5,16}\\ {\mathsf{\Phi }}_{26,16}\\ {\mathsf{\Phi }}_{18,16}\end{array}\end{array}\right)+\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Omega }}_{5,23}\\ {\mathsf{\Omega }}_{7,23}\\ {\mathsf{\Omega }}_{8,23}\\ {\mathsf{\Omega }}_{10,23}\end{array}\end{array}\right),\hfill \\ \hfill & {\mathsf{\Psi }}_{20,6}={\mathsf{\Phi }}_{16,27}+{\mathsf{\Psi }}_{16,4}+{\mathsf{\Omega }}_{23,6},\hfill \\ \hfill & {\mathsf{\Psi }}_{6,20}={\mathsf{\Phi }}_{27,16}+{\mathsf{\Psi }}_{4,16}+{\mathsf{\Omega }}_{6,23},\hfill \end{array}\right.\end{array}$$

(41)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Psi }}_{18,3}\\ {\mathsf{\Psi }}_{18,8}\\ {\mathsf{\Psi }}_{18,14}\\ {\mathsf{\Psi }}_{18,16}\end{array}\end{array}\right)}^{\mathsf{T}}={\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Phi }}_{9,28}\\ {\mathsf{\Phi }}_{9,26}\\ {\mathsf{\Phi }}_{9,25}\\ {\mathsf{\Phi }}_{9,23}\end{array}\end{array}\right)}^{\mathsf{T}}+{\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Omega }}_{21,2}\\ {\mathsf{\Omega }}_{21,8}\\ {\mathsf{\Omega }}_{21,14}\\ {\mathsf{\Omega }}_{21,19}\end{array}\end{array}\right)}^{\mathsf{T}},\hfill \\ \hfill & \left(\begin{array}{c}\begin{array}{c}{\mathsf{\Psi }}_{3,18}\\ {\mathsf{\Psi }}_{8,18}\\ {\mathsf{\Psi }}_{14,18}\\ {\mathsf{\Psi }}_{16,18}\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Phi }}_{28,9}\\ {\mathsf{\Phi }}_{26,9}\\ {\mathsf{\Phi }}_{25,9}\\ {\mathsf{\Phi }}_{23,9}\end{array}\end{array}\right)+\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Omega }}_{2,21}\\ {\mathsf{\Omega }}_{8,21}\\ {\mathsf{\Omega }}_{14,21}\\ {\mathsf{\Omega }}_{19,21}\end{array}\end{array}\right),\hfill \\ \hfill & {\mathsf{\Psi }}_{18,6}={\mathsf{\Phi }}_{9,27}+{\mathsf{\Omega }}_{21,16}+{\mathsf{\Omega }}_{21,6},\hfill \\ \hfill & {\mathsf{\Psi }}_{6,18}={\mathsf{\Phi }}_{27,9}+{\mathsf{\Omega }}_{16,21}+{\mathsf{\Omega }}_{6,21},\hfill \end{array}\right.\end{array}$$

(42)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \left(\begin{array}{c}\begin{array}{c}{\mathsf{\Psi }}_{8,15}\\ {\mathsf{\Psi }}_{9,15}\\ {\mathsf{\Psi }}_{10,15}\\ {\mathsf{\Psi }}_{16,15}\\ {\mathsf{\Psi }}_{17,15}\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Phi }}_{26,3}\\ {\mathsf{\Phi }}_{24,3}\\ {\mathsf{\Phi }}_{18,3}\\ {\mathsf{\Phi }}_{23,3}\\ {\mathsf{\Phi }}_{17,3}\end{array}\end{array}\right)+\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Omega }}_{8,15}\\ {\mathsf{\Omega }}_{9,15}\\ {\mathsf{\Omega }}_{10,15}\\ {\mathsf{\Omega }}_{19,15}\\ {\mathsf{\Omega }}_{20,15}\end{array}\end{array}\right),\hfill \\ \hfill & {\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Psi }}_{9,18}\\ {\mathsf{\Psi }}_{9,20}\\ {\mathsf{\Psi }}_{9,21}\end{array}\end{array}\right)}^{\mathsf{T}}={\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Phi }}_{24,9}\\ {\mathsf{\Phi }}_{24,16}\\ {\mathsf{\Phi }}_{24,20}\end{array}\end{array}\right)}^{\mathsf{T}}+{\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Omega }}_{9,21}\\ {\mathsf{\Omega }}_{9,23}\\ {\mathsf{\Omega }}_{9,27}\end{array}\end{array}\right)}^{\mathsf{T}},\hfill \\ \hfill & {\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Psi }}_{15,8}\\ {\mathsf{\Psi }}_{15,9}\\ {\mathsf{\Psi }}_{15,10}\\ {\mathsf{\Psi }}_{15,16}\\ {\mathsf{\Psi }}_{15,17}\end{array}\end{array}\right)}^{\mathsf{T}}={\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Phi }}_{3,26}\\ {\mathsf{\Phi }}_{3,24}\\ {\mathsf{\Phi }}_{3,18}\\ {\mathsf{\Phi }}_{3,23}\\ {\mathsf{\Phi }}_{3,17}\end{array}\end{array}\right)}^{\mathsf{T}}+{\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Omega }}_{15,8}\\ {\mathsf{\Omega }}_{15,9}\\ {\mathsf{\Omega }}_{15,10}\\ {\mathsf{\Omega }}_{15,19}\\ {\mathsf{\Omega }}_{15,20}\end{array}\end{array}\right)}^{\mathsf{T}},\hfill \\ \hfill & \left(\begin{array}{c}\begin{array}{c}{\mathsf{\Psi }}_{18,9}\\ {\mathsf{\Psi }}_{20,9}\\ {\mathsf{\Psi }}_{21,9}\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Phi }}_{9,24}\\ {\mathsf{\Phi }}_{16,24}\\ {\mathsf{\Phi }}_{20,24}\end{array}\end{array}\right)+\left(\begin{array}{c}\begin{array}{c}{\mathsf{\Omega }}_{21,9}\\ {\mathsf{\Omega }}_{23,9}\\ {\mathsf{\Omega }}_{27,9}\end{array}\end{array}\right),\hfill \\ \hfill & {\mathsf{\Psi }}_{8,17}={\mathsf{\Phi }}_{26,17}+{\mathsf{\Phi }}_{26,8}+{\mathsf{\Omega }}_{8,24}+{\mathsf{\Omega }}_{8,20},\hfill \\ \hfill & {\mathsf{\Psi }}_{9,17}={\mathsf{\Phi }}_{24,17}+{\mathsf{\Phi }}_{24,8}+{\mathsf{\Omega }}_{9,24}+{\mathsf{\Omega }}_{9,20},\hfill \\ \hfill & {\mathsf{\Psi }}_{10,17}={\mathsf{\Phi }}_{18,17}+{\mathsf{\Phi }}_{18,8}+{\mathsf{\Omega }}_{10,24}+{\mathsf{\Omega }}_{10,20},\hfill \\ \hfill & {\mathsf{\Psi }}_{17,17}={\mathsf{\Phi }}_{17,17}+{\mathsf{\Phi }}_{17,8}+{\mathsf{\Omega }}_{20,24}+{\mathsf{\Omega }}_{20,20},\hfill \\ \hfill & {\mathsf{\Psi }}_{17,8}={\mathsf{\Phi }}_{17,26}+{\mathsf{\Phi }}_{8,26}+{\mathsf{\Omega }}_{24,8}+{\mathsf{\Omega }}_{20,8},\hfill \\ \hfill & {\mathsf{\Psi }}_{17,9}={\mathsf{\Phi }}_{17,24}+{\mathsf{\Phi }}_{8,24}+{\mathsf{\Omega }}_{24,9}+{\mathsf{\Omega }}_{20,9},\hfill \\ \hfill & {\mathsf{\Psi }}_{17,10}={\mathsf{\Phi }}_{17,18}+{\mathsf{\Phi }}_{8,18}+{\mathsf{\Omega }}_{24,10}+{\mathsf{\Omega }}_{20,10},\hfill \end{array}\right.\end{array}$$

(43)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\mathsf{\Psi }}_{18,13}={\mathsf{\Phi }}_{9,24}+{\mathsf{\Omega }}_{21,13},\hfill \\ \hfill & {\mathsf{\Psi }}_{13,18}={\mathsf{\Phi }}_{24,9}+{\mathsf{\Omega }}_{13,21},\hfill \\ \hfill & {\mathsf{\Psi }}_{16,13}={\mathsf{\Phi }}_{23,24}+{\mathsf{\Phi }}_{7,24}+{\mathsf{\Phi }}_{23,1}+{\mathsf{\Omega }}_{19,13},\hfill \\ \hfill & {\mathsf{\Psi }}_{17,13}={\mathsf{\Phi }}_{17,24}+{\mathsf{\Phi }}_{8,24}+{\mathsf{\Phi }}_{17,1}+{\mathsf{\Omega }}_{20,13},\hfill \\ \hfill & {\mathsf{\Psi }}_{16,14}={\mathsf{\Phi }}_{23,25}+{\mathsf{\Phi }}_{7,25}+{\mathsf{\Phi }}_{23,2}+{\mathsf{\Omega }}_{19,14},\hfill \\ \hfill & {\mathsf{\Psi }}_{17,14}={\mathsf{\Phi }}_{17,25}+{\mathsf{\Phi }}_{8,25}+{\mathsf{\Phi }}_{17,2}+{\mathsf{\Omega }}_{20,14},\hfill \\ \hfill & {\mathsf{\Psi }}_{16,16}={\mathsf{\Phi }}_{23,23}+{\mathsf{\Phi }}_{7,23}+{\mathsf{\Phi }}_{23,7}+{\mathsf{\Omega }}_{19,19},\hfill \\ \hfill & {\mathsf{\Psi }}_{17,16}={\mathsf{\Phi }}_{17,23}+{\mathsf{\Phi }}_{8,23}+{\mathsf{\Phi }}_{17,7}+{\mathsf{\Omega }}_{20,19},\hfill \\ \hfill & {\mathsf{\Psi }}_{13,16}={\mathsf{\Phi }}_{24,23}+{\mathsf{\Phi }}_{1,23}+{\mathsf{\Phi }}_{24,7}+{\mathsf{\Omega }}_{13,19},\hfill \\ \hfill & {\mathsf{\Psi }}_{14,16}={\mathsf{\Phi }}_{25,23}+{\mathsf{\Phi }}_{2,23}+{\mathsf{\Phi }}_{25,7}+{\mathsf{\Omega }}_{14,19},\hfill \\ \hfill & {\mathsf{\Psi }}_{13,17}={\mathsf{\Phi }}_{24,17}+{\mathsf{\Phi }}_{1,17}+{\mathsf{\Phi }}_{24,8}+{\mathsf{\Omega }}_{13,20},\hfill \\ \hfill & {\mathsf{\Psi }}_{14,17}={\mathsf{\Phi }}_{25,17}+{\mathsf{\Phi }}_{2,17}+{\mathsf{\Phi }}_{25,8}+{\mathsf{\Omega }}_{14,20},\hfill \\ \hfill & {\mathsf{\Psi }}_{16,17}={\mathsf{\Phi }}_{23,17}+{\mathsf{\Phi }}_{7,17}+{\mathsf{\Phi }}_{23,8}+{\mathsf{\Omega }}_{19,20},\hfill \end{array}\right.\end{array}$$

(44)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\mathsf{\Psi }}_{10,7}={\mathsf{\Phi }}_{18,5}+{\mathsf{\Omega }}_{10,17}+{\mathsf{\Omega }}_{25,7}+{\mathsf{\Omega }}_{10,7},\hfill \\ \hfill & {\mathsf{\Psi }}_{17,7}={\mathsf{\Phi }}_{17,5}+{\mathsf{\Omega }}_{20,17}+{\mathsf{\Omega }}_{24,7}+{\mathsf{\Omega }}_{20,7},\hfill \\ \hfill & {\mathsf{\Psi }}_{7,10}={\mathsf{\Phi }}_{5,18}+{\mathsf{\Omega }}_{7,25}+{\mathsf{\Omega }}_{17,10}+{\mathsf{\Omega }}_{7,10},\hfill \\ \hfill & {\mathsf{\Psi }}_{10,10}={\mathsf{\Phi }}_{18,18}+{\mathsf{\Omega }}_{10,25}+{\mathsf{\Omega }}_{25,10}+{\mathsf{\Omega }}_{10,10},\hfill \\ \hfill & {\mathsf{\Psi }}_{7,17}={\mathsf{\Phi }}_{5,17}+{\mathsf{\Omega }}_{7,24}+{\mathsf{\Omega }}_{17,20}+{\mathsf{\Omega }}_{7,20},\hfill \end{array}\right.\end{array}$$

(45)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\mathsf{\Psi }}_{17,6}={\mathsf{\Phi }}_{17,27}+{\mathsf{\Phi }}_{8,27}+{\mathsf{\Phi }}_{17,4}+{\mathsf{\Omega }}_{20,16}+{\mathsf{\Omega }}_{24,6}+{\mathsf{\Omega }}_{20,6},\hfill \\ \hfill & {\mathsf{\Psi }}_{6,17}={\mathsf{\Phi }}_{27,17}+{\mathsf{\Phi }}_{27,8}+{\mathsf{\Phi }}_{4,17}+{\mathsf{\Omega }}_{16,20}+{\mathsf{\Omega }}_{6,24}+{\mathsf{\Omega }}_{6,20}.\hfill \end{array}\right.\end{array}$$

(46)

Now we want to prove that (8)–(23) ⇒ (31)–(46). First, we show that (8) ⇒ (31). Upon computations, we obtain the following:

$$\begin{array}{c}\hfill r\left(\begin{array}{ccc}A& B& \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{cc}A& B\end{array}\right)⟺r\left(\begin{array}{ccc}{S}_a& S_b& {\left({\mathsf{\Phi }}_{i,j}\right)}_{30\times 30}\end{array}\right)=r\left(\begin{array}{cc}{S}_a& S_b\end{array}\right)\end{array}$$

$$\begin{array}{c}\hfill \Rightarrow \left(\begin{array}{cccc}{\mathsf{\Phi }}_{30,1}& {\mathsf{\Phi }}_{30,2}& \dots & {\mathsf{\Phi }}_{30,29}\end{array}\right)=0,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}F\\ G\\ \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{c}F\\ G\end{array}\right)⟺r\left(\begin{array}{c}{S}_f\\ S_g\\ {\left({\mathsf{\Phi }}_{i,j}\right)}_{30\times 30}\end{array}\right)=r\left(\begin{array}{c}{S}_f\\ S_g\end{array}\right)\Rightarrow \left(\begin{array}{c}{\mathsf{\Phi }}_{1,30}\\ {\mathsf{\Phi }}_{2,30}\\ ⋮\\ {\mathsf{\Phi }}_{30,30}\end{array}\right)=0,\end{array}$$

Similarly, we can prove that (9) ⇒ (32); (10) ⇒ (33); (11) ⇒ (34); (12) ⇒ (35); (13) ⇒ (36); (14) ⇒ (37); (15) ⇒ (38); (16) ⇒ (39); (17) ⇒ (40); (18) ⇒ (41); (19) ⇒ (42); (20) ⇒ (43); (21) ⇒ (44); (22) ⇒ (45); (23) ⇒ (46). □

Now set matrices A and H to be zero matrices, and system (4) takes the following form:

$$\begin{array}{c}\hfill \left\{\begin{array}{cccc}BYG& & =& \mathsf{\Phi },\\ CYH& & =& \mathsf{\Psi },\\ DYJ& +EZK& =& \mathsf{\Omega }.\end{array}\right.\end{array}$$

(47)

He et al. [3] presented a necessary and sufficient condition for the solvability of system (47), which can also be derived using Theorem 1.

Corollary 1 

([3]). The system (47) is consistent if and only if the following rank equalities hold:

$$\begin{array}{c}\hfill r\left(\begin{array}{cc}B& \mathsf{\Phi }\end{array}\right)=r\left(B\right),r\left(\begin{array}{c}G\\ \mathsf{\Phi }\end{array}\right)=r\left(G\right),r\left(\begin{array}{cc}C& \mathsf{\Psi }\end{array}\right)=r\left(C\right),r\left(\begin{array}{c}H\\ \mathsf{\Psi }\end{array}\right)=r\left(H\right),\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{ccc}D& E& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cc}D& E\end{array}\right),r\left(\begin{array}{c}J\\ K\\ \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{c}J\\ K\end{array}\right),r\left(\begin{array}{cc}0& K\\ D& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cc}0& K\\ D& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{cc}0& J\\ E& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cc}0& J\\ E& 0\end{array}\right),r\left(\begin{array}{ccc}0& G& H\\ B& \mathsf{\Phi }& 0\\ C& 0& -\mathsf{\Psi }\end{array}\right)=r\left(\begin{array}{cc}G& H\end{array}\right)+r\left(\begin{array}{c}B\\ C\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{cccc}0& 0& J& G\\ D& E& -\mathsf{\Omega }& 0\\ B& 0& 0& \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{cc}J& G\end{array}\right)+r\left(\begin{array}{cc}D& E\\ B& 0\end{array}\right),r\left(\begin{array}{ccc}0& J& G\\ 0& K& 0\\ D& -\mathsf{\Omega }& 0\\ B& 0& \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{cc}J& G\\ K& 0\end{array}\right)+r\left(\begin{array}{c}D\\ B\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{cccc}0& 0& J& H\\ D& E& -\mathsf{\Omega }& 0\\ C& 0& 0& \mathsf{\Psi }\end{array}\right)=r\left(\begin{array}{cc}J& H\end{array}\right)+r\left(\begin{array}{cc}D& E\\ C& 0\end{array}\right),r\left(\begin{array}{ccc}0& J& H\\ 0& K& 0\\ D& -\mathsf{\Omega }& 0\\ C& 0& \mathsf{\Psi }\end{array}\right)=r\left(\begin{array}{cc}J& H\\ K& 0\end{array}\right)+r\left(\begin{array}{c}D\\ C\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{ccccc}0& 0& J& G& H\\ 0& 0& K& 0& 0\\ D& D& -\mathsf{\Omega }& 0& 0\\ B& 0& 0& \mathsf{\Phi }& 0\\ 0& C& 0& 0& \mathsf{\Psi }\end{array}\right)=r\left(\begin{array}{ccc}J& G& H\\ K& 0& 0\end{array}\right)+r\left(\begin{array}{cc}D& D\\ B& 0\\ 0& C\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{ccccc}0& 0& J& G& 0\\ 0& 0& J& 0& H\\ D& E& -\mathsf{\Omega }& 0& 0\\ B& 0& 0& \mathsf{\Phi }& 0\\ C& 0& 0& 0& \mathsf{\Psi }\end{array}\right)=r\left(\begin{array}{ccc}J& G& 0\\ J& 0& H\end{array}\right)+r\left(\begin{array}{cc}D& E\\ B& 0\\ C& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{cccccc}0& 0& 0& J& G& H\\ D& D& E& -\mathsf{\Omega }& 0& 0\\ B& 0& 0& 0& \mathsf{\Phi }& 0\\ 0& C& 0& 0& 0& \mathsf{\Psi }\end{array}\right)=r\left(\begin{array}{ccc}J& G& H\end{array}\right)+r\left(\begin{array}{ccc}D& D& E\\ B& 0& 0\\ 0& C& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{cccc}0& J& G& 0\\ 0& J& 0& H\\ 0& K& 0& 0\\ D& -\mathsf{\Omega }& 0& 0\\ B& 0& \mathsf{\Phi }& 0\\ C& 0& 0& \mathsf{\Psi }\end{array}\right)=r\left(\begin{array}{ccc}J& G& 0\\ 0& J& 0& H\\ 0& K& 0& 0\end{array}\right)+r\left(\begin{array}{c}D\\ B\\ C\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& K\\ 0& 0& 0& H& 0& J& 0\\ 0& 0& 0& 0& G& J& J\\ 0& B& 0& 0& -\mathsf{\Phi }& 0& 0\\ 0& 0& C& \mathsf{\Psi }& 0& 0& 0\\ 0& D& D& 0& 0& 0& \mathsf{\Omega }\\ E& 0& D& 0& 0& -\mathsf{\Omega }& 0\end{array}\right)=r\left(\begin{array}{cccc}0& 0& 0& K\\ H& 0& J& 0\\ 0& G& J& J\end{array}\right)+r\left(\begin{array}{ccc}0& B& 0\\ 0& 0& C\\ 0& D& D\\ E& 0& D\end{array}\right).\end{array}$$

4. Examples

This section illustrates the results from Section 3 with numerical examples.

Example 1. 

For the system (4)

$$\begin{array}{c}\hfill \left\{\begin{array}{ccccc}AXF+& BYG& & =& \mathsf{\Phi },\\ & CYH& & =& \mathsf{\Psi },\\ & DYJ& +EZK& =& \mathsf{\Omega }.\end{array}\right.\end{array}$$

Let

$$\begin{array}{c}A=\left(\begin{array}{c}\mathbf{i}\\ 0\end{array}\right),B=\left(\begin{array}{ccc}1& \mathbf{i}& 0\\ \mathbf{j}& 0& \mathbf{k}\end{array}\right),C=\left(\begin{array}{ccc}2\mathbf{j}& -\mathbf{k}& \mathbf{k}\\ 0& \mathbf{i}& \mathbf{i}\\ -\mathbf{k}& 0& \mathbf{j}\end{array}\right),D=\left(\begin{array}{ccc}-\mathbf{j}& 1& 1\\ 0& 0& \mathbf{i}\end{array}\right),E=\left(\begin{array}{cc}0& 0\\ 0& \mathbf{i}\end{array}\right),\hfill \\ F=\left(\begin{array}{cc}\mathbf{j}& 0\end{array}\right),G=\left(\begin{array}{cc}0& 1\\ \mathbf{k}& -\mathbf{i}\\ \mathbf{i}& 0\end{array}\right),H=\left(\begin{array}{ccc}\mathbf{i}& -\mathbf{j}& 0\\ 1& \mathbf{k}& \mathbf{i}\\ 0& \mathbf{j}& \mathbf{j}\end{array}\right),J=\left(\begin{array}{cc}\mathbf{i}& -\mathbf{k}\\ 2& 0\\ 0& 1\end{array}\right),K=\left(\begin{array}{cc}\mathbf{k}& 0\end{array}\right),\hfill \\ \mathsf{\Phi }=\left(\begin{array}{cc}1-2\mathbf{i}& \mathbf{i}+\mathbf{j}\\ \mathbf{j}& -1-\mathbf{k}\end{array}\right),\mathsf{\Psi }=\left(\begin{array}{ccc}-2\mathbf{i}-2\mathbf{j}& -2\mathbf{i}+2\mathbf{j}& 2\\ 0& 0& 0\\ -1+\mathbf{k}& -1-\mathbf{k}& -\mathbf{i}\end{array}\right),\mathsf{\Omega }=\left(\begin{array}{cc}2\mathbf{i}+\mathbf{j}& 1\\ 0& 0\end{array}\right).\hfill \end{array}$$

By a direct calculation, we have the following:

$$\begin{array}{c}\hfill r\left(\begin{array}{ccc}A& B& \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{cc}A& B\end{array}\right)=2,r\left(\begin{array}{c}F\\ G\\ \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{c}F\\ G\end{array}\right)=2,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{cc}C& \mathsf{\Psi }\end{array}\right)=r\left(C\right)=2,r\left(\begin{array}{c}H\\ \mathsf{\Psi }\end{array}\right)=r\left(H\right)=3,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{ccc}D& E& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cc}D& E\end{array}\right)=2,r\left(\begin{array}{c}J\\ K\\ \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{c}J\\ K\end{array}\right)=2,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{cc}0& G\\ A& \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{cc}0& G\\ A& 0\end{array}\right)=3,r\left(\begin{array}{cc}0& F\\ B& \mathsf{\Phi }\end{array}\right)=r\left(\begin{array}{cc}0& F\\ B& 0\end{array}\right)=3,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{cc}0& K\\ D& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cc}0& K\\ D& 0\end{array}\right)=3,r\left(\begin{array}{cc}0& J\\ E& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cc}0& J\\ E& 0\end{array}\right)=3,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{cccc}0& 0& G& H\\ A& B& -\mathsf{\Phi }& 0\\ 0& C& 0& \mathsf{\Psi }\end{array}\right)=r\left(\begin{array}{cccc}0& 0& G& H\\ A& B& 0& 0\\ 0& C& 0& 0\end{array}\right)=6,r\left(\begin{array}{ccc}0& F& 0\\ 0& G& H\\ B& -\mathsf{\Phi }& 0\\ C& 0& \mathsf{\Psi }\end{array}\right)=r\left(\begin{array}{ccc}0& F& 0\\ 0& G& H\\ B& 0& 0\\ C& 0& 0\end{array}\right)=6,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{cccc}0& 0& H& J\\ C& 0& -\mathsf{\Psi }& 0\\ D& E& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{cccc}0& 0& H& J\\ C& 0& 0& 0\\ D& E& 0& 0\end{array}\right)=7,r\left(\begin{array}{ccc}0& H& J\\ 0& 0& K\\ C& -\mathsf{\Psi }& 0\\ D& 0& \mathsf{\Omega }\end{array}\right)=r\left(\begin{array}{ccc}0& H& J\\ 0& 0& K\\ C& 0& 0\\ D& 0& 0\end{array}\right)=7,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{cccc}0& 0& G& J\\ 0& 0& 0& K\\ A& B& -\mathsf{\Phi }& 0\\ 0& D& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cccc}0& 0& G& J\\ 0& 0& 0& K\\ A& B& 0& 0\\ 0& D& 0& 0\end{array}\end{array}\right)=8,r\left(\begin{array}{c}\begin{array}{cccc}0& 0& F& 0\\ 0& 0& G& J\\ B& 0& -\mathsf{\Phi }& 0\\ D& E& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cccc}0& 0& F& 0\\ 0& 0& G& J\\ B& 0& 0& 0\\ D& E& 0& 0\end{array}\end{array}\right)=8,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& 0& G& J\\ A& B& 0& -\mathsf{\Phi }& 0\\ 0& D& E& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& 0& G& J\\ A& B& 0& 0& 0\\ 0& D& E& 0& 0\end{array}\end{array}\right)=7,r\left(\begin{array}{c}\begin{array}{ccc}0& F& 0\\ 0& G& J\\ 0& 0& K\\ B& -\mathsf{\Phi }& 0\\ D& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccc}0& F& 0\\ 0& G& J\\ 0& 0& K\\ B& 0& 0\\ D& 0& 0\end{array}\end{array}\right)=7,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{ccccccc}0& 0& 0& 0& G& H& J\\ A& B& 0& 0& \mathsf{\Phi }& 0& 0\\ 0& C& C& 0& 0& -\mathsf{\Psi }& 0\\ 0& 0& D& E& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccccccc}0& 0& 0& 0& G& H& J\\ A& B& 0& 0& 0& 0& 0\\ 0& C& C& 0& 0& 0& 0\\ 0& 0& D& E& 0& 0& 0\end{array}\end{array}\right)=9,r\left(\begin{array}{c}\begin{array}{cccc}0& F& 0& 0\\ 0& G& H& 0\\ 0& 0& H& J\\ 0& 0& 0& K\\ B& \mathsf{\Phi }& 0& 0\\ C& 0& -\mathsf{\Psi }& 0\\ D& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cccc}0& F& 0& 0\\ 0& G& H& 0\\ 0& 0& H& J\\ 0& 0& 0& K\\ B& 0& 0& 0\\ C& 0& 0& 0\\ D& 0& 0& 0\end{array}\end{array}\right)=10,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{cccccc}0& 0& 0& G& H& J\\ 0& 0& 0& 0& 0& K\\ A& B& 0& \mathsf{\Phi }& 0& 0\\ 0& C& C& 0& -\mathsf{\Psi }& 0\\ 0& 0& D& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cccccc}0& 0& 0& G& H& J\\ 0& 0& 0& 0& 0& K\\ A& B& 0& 0& 0& 0\\ 0& C& C& 0& 0& 0\\ 0& 0& D& 0& 0& 0\end{array}\end{array}\right)=10,r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& F& 0& 0\\ 0& 0& G& H& 0\\ 0& 0& 0& H& J\\ B& 0& \mathsf{\Phi }& 0& 0\\ C& 0& 0& -\mathsf{\Psi }& 0\\ D& E& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& F& 0& 0\\ 0& 0& G& H& 0\\ 0& 0& 0& H& J\\ B& 0& 0& 0& 0\\ C& 0& 0& 0& 0\\ D& E& 0& 0& 0\end{array}\end{array}\right)=11,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{cccccc}0& 0& 0& F& 0& 0\\ 0& 0& 0& G& H& J\\ B& 0& 0& \mathsf{\Phi }& 0& 0\\ C& C& 0& 0& -\mathsf{\Psi }& 0\\ 0& D& E& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cccccc}0& 0& 0& F& 0& 0\\ 0& 0& 0& G& H& J\\ B& 0& 0& 0& 0& 0\\ C& C& 0& 0& 0& 0\\ 0& D& E& 0& 0& 0\end{array}\end{array}\right)=10,r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& G& H& 0\\ 0& 0& 0& H& J\\ 0& 0& 0& 0& K\\ A& B& \mathsf{\Phi }& 0& 0\\ 0& C& 0& -\mathsf{\Psi }& 0\\ 0& D& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& G& H& 0\\ 0& 0& 0& H& J\\ 0& 0& 0& 0& K\\ A& B& 0& 0& 0\\ 0& C& 0& 0& 0\\ 0& D& 0& 0& 0\end{array}\end{array}\right)=11,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{cccccc}0& 0& 0& G& H& 0\\ 0& 0& 0& 0& H& J\\ A& B& 0& \mathsf{\Phi }& 0& 0\\ 0& C& 0& 0& -\mathsf{\Psi }& 0\\ 0& D& E& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cccccc}0& 0& 0& G& H& 0\\ 0& 0& 0& 0& H& J\\ A& B& 0& 0& 0& 0\\ 0& C& 0& 0& 0& 0\\ 0& D& E& 0& 0& 0\end{array}\end{array}\right)=11,r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& F& 0& 0\\ 0& 0& G& H& J\\ 0& 0& 0& 0& K\\ B& 0& \mathsf{\Phi }& 0& 0\\ C& C& 0& -\mathsf{\Psi }& 0\\ 0& D& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccccc}0& 0& F& 0& 0\\ 0& 0& G& H& J\\ 0& 0& 0& 0& K\\ B& 0& 0& 0& 0\\ C& C& 0& 0& 0\\ 0& D& 0& 0& 0\end{array}\end{array}\right)=10,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{cccccccc}0& 0& 0& 0& 0& F& 0& 0\\ 0& 0& 0& 0& G& 0& H& 0\\ 0& 0& 0& 0& 0& G& H& J\\ A& B& 0& 0& \mathsf{\Phi }& 0& 0& 0\\ 0& 0& B& 0& 0& -\mathsf{\Phi }& 0& 0\\ 0& C& 0& 0& 0& 0& -\mathsf{\Psi }& 0\\ 0& D& D& E& 0& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cccccccc}0& 0& 0& 0& 0& F& 0& 0\\ 0& 0& 0& 0& G& 0& H& 0\\ 0& 0& 0& 0& 0& G& H& J\\ A& B& 0& 0& 0& 0& 0& 0\\ 0& 0& B& 0& 0& 0& 0& 0\\ 0& C& 0& 0& 0& 0& 0& 0\\ 0& D& D& E& 0& 0& 0& 0\end{array}\end{array}\right)=14,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{ccccccc}0& 0& 0& F& 0& 0& 0\\ 0& 0& 0& G& 0& H& J\\ 0& 0& 0& 0& G& 0& J\\ 0& 0& 0& 0& 0& 0& K\\ 0& B& 0& \mathsf{\Phi }& 0& 0& 0\\ A& 0& B& 0& -\mathsf{\Phi }& 0& 0\\ 0& C& C& 0& 0& -\mathsf{\Psi }& 0\\ 0& 0& D& 0& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccccccc}0& 0& 0& F& 0& 0& 0\\ 0& 0& 0& G& 0& H& J\\ 0& 0& 0& 0& G& 0& J\\ 0& 0& 0& 0& 0& 0& K\\ 0& B& 0& 0& 0& 0& 0\\ A& 0& B& 0& 0& 0& 0\\ 0& C& C& 0& 0& 0& 0\\ 0& 0& D& 0& 0& 0& 0\end{array}\end{array}\right)=14,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{cccccccc}0& 0& 0& 0& G& H& J& 0\\ 0& 0& 0& 0& 0& H& 0& J\\ 0& 0& 0& 0& 0& 0& K& 0\\ A& B& B& 0& \mathsf{\Phi }& 0& 0& 0\\ 0& C& 0& 0& 0& -\mathsf{\Psi }& 0& 0\\ 0& 0& D& 0& 0& 0& -\mathsf{\Omega }& 0\\ 0& D& 0& E& 0& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cccccccc}0& 0& 0& 0& G& H& J& 0\\ 0& 0& 0& 0& 0& H& 0& J\\ 0& 0& 0& 0& 0& 0& K& 0\\ A& B& B& 0& 0& 0& 0& 0\\ 0& C& 0& 0& 0& 0& 0& 0\\ 0& 0& D& 0& 0& 0& 0& 0\\ 0& D& 0& E& 0& 0& 0& 0\end{array}\end{array}\right)=15,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{ccccccc}0& 0& 0& F& 0& 0& 0\\ 0& 0& 0& G& H& 0& J\\ 0& 0& 0& G& 0& J& 0\\ 0& 0& 0& 0& 0& 0& K\\ B& 0& 0& \mathsf{\Phi }& 0& 0& 0\\ C& C& 0& 0& -\mathsf{\Psi }& 0& 0\\ D& 0& E& 0& 0& -\mathsf{\Omega }& 0\\ 0& D& 0& 0& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccccccc}0& 0& 0& F& 0& 0& 0\\ 0& 0& 0& G& H& 0& J\\ 0& 0& 0& G& 0& J& 0\\ 0& 0& 0& 0& 0& 0& K\\ B& 0& 0& 0& 0& 0& 0\\ C& C& 0& 0& 0& 0& 0\\ D& 0& E& 0& 0& 0& 0\\ 0& D& 0& 0& 0& 0& 0\end{array}\end{array}\right)=15,\end{array}$$

$$\begin{array}{c}\hfill r\left(\begin{array}{c}\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& F& 0& 0& 0\\ 0& 0& 0& 0& 0& G& 0& H& J& 0\\ 0& 0& 0& 0& 0& 0& G& H& 0& J\\ 0& 0& 0& 0& 0& 0& 0& 0& K& 0\\ A& B& 0& B& 0& \mathsf{\Phi }& 0& 0& 0& 0\\ 0& 0& B& 0& 0& 0& -\mathsf{\Phi }& 0& 0& 0\\ 0& C& 0& 0& 0& 0& 0& -\mathsf{\Psi }& 0& 0\\ 0& 0& 0& D& 0& 0& 0& 0& -\mathsf{\Omega }& 0\\ 0& D& D& 0& E& 0& 0& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{cccccccccc}0& 0& 0& 0& 0& 0& F& 0& 0& 0\\ 0& 0& 0& 0& 0& G& 0& H& J& 0\\ 0& 0& 0& 0& 0& 0& G& H& 0& J\\ 0& 0& 0& 0& 0& 0& 0& 0& K& 0\\ A& B& 0& B& 0& 0& 0& 0& 0& 0\\ 0& 0& B& 0& 0& 0& 0& 0& 0& 0\\ 0& C& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& D& 0& 0& 0& 0& 0& 0\\ 0& D& D& 0& E& 0& 0& 0& 0& 0\end{array}\end{array}\right)=18,\\ \hfill r\left(\begin{array}{c}\begin{array}{ccccccccc}0& 0& 0& 0& F& 0& 0& 0& 0\\ 0& 0& 0& 0& G& 0& H& 0& J\\ 0& 0& 0& 0& 0& G& 0& 0& J\\ 0& 0& 0& 0& G& 0& 0& J& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& K\\ 0& B& 0& 0& \mathsf{\Phi }& 0& 0& 0& 0\\ A& 0& B& 0& 0& -\mathsf{\Phi }& 0& 0& 0\\ 0& C& C& 0& 0& 0& -\mathsf{\Psi }& 0& 0\\ 0& D& 0& E& 0& 0& 0& -\mathsf{\Omega }& 0\\ 0& 0& D& 0& 0& 0& 0& 0& \mathsf{\Omega }\end{array}\end{array}\right)=r\left(\begin{array}{c}\begin{array}{ccccccccc}0& 0& 0& 0& F& 0& 0& 0& 0\\ 0& 0& 0& 0& G& 0& H& 0& J\\ 0& 0& 0& 0& 0& G& 0& 0& J\\ 0& 0& 0& 0& G& 0& 0& J& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& K\\ 0& B& 0& 0& 0& 0& 0& 0& 0\\ A& 0& B& 0& 0& 0& 0& 0& 0\\ 0& C& C& 0& 0& 0& 0& 0& 0\\ 0& D& 0& E& 0& 0& 0& 0& 0\\ 0& 0& D& 0& 0& 0& 0& 0& 0\end{array}\end{array}\right)=19.\end{array}$$

All the rank equalities in (8)–(23) hold. Therefore, the system (4) is consistent. Based on the proposed simultaneous decomposition, we obtain the following:

$$\begin{array}{c}\hfill P_1=\left(\begin{array}{cc}-\mathbf{i}& -1-\mathbf{k}\\ 0& -\mathbf{j}\end{array}\right), P_2=\left(\begin{array}{ccc}\mathbf{k}& 0& -\mathbf{i}-2\mathbf{j}\\ \mathbf{j}& 1& 2\mathbf{k}\\ 0& 0& \mathbf{k}\end{array}\right), P_3=\left(\begin{array}{cc}-\mathbf{k}& 0\\ 1& \mathbf{j}\end{array}\right), \end{array}$$

$$\begin{array}{c}\hfill Q_1=1, Q_2=\left(\begin{array}{ccc}1& -\mathbf{j}& \mathbf{i}\\ \mathbf{j}& 1-\mathbf{k}& -1\\ 0& \mathbf{k}& 1\end{array}\right), Q_3=\left(\begin{array}{cc}1& 0\\ 0& \mathbf{k}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill \widetilde{P_1}=\mathbf{i}, \widetilde{P_2}=\left(\begin{array}{ccc}2& -\mathbf{i}& 2\mathbf{k}\\ 0.5\mathbf{i}& 0.5& -0.5\mathbf{j}\\ \mathbf{k}& -\mathbf{j}& 1\end{array}\right), \widetilde{P_3}=-0.5\mathbf{k},\end{array}$$

$$\begin{array}{c}\hfill \widetilde{Q_1}=\left(\begin{array}{cc}-\mathbf{k}& 0\\ 3\mathbf{i}& 1\end{array}\right), \widetilde{Q_2}=\left(\begin{array}{ccc}\mathbf{i}& 0.5-0.5\mathbf{i}-0.5\mathbf{j}& 3-\mathbf{j}-\mathbf{k}\\ 0& -0.5\mathbf{j}+0.5\mathbf{k}& 1+\mathbf{i}\\ 0& -0.5\mathbf{k}& -\mathbf{i}\end{array}\right), \widetilde{Q_3}=\left(\begin{array}{cc}0& 2\\ 0.5& \mathbf{j}\end{array}\right),\end{array}$$

such that

$$\begin{array}{c}\hfill S_a=P_{1}A{Q}_1=\left(\begin{array}{c}1\\ 0\end{array}\right), S_b=P_{1}B{Q}_2=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill S_c=P_{2}C{Q}_2=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 1& 0& 0\end{array}\right), S_d=P_{3}D{Q}_2=\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\end{array}\right), S_e=P_{3}E{Q}_3=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill S_f=\widetilde{P_1}F\widetilde{Q_1}=\left(\begin{array}{cc}1& 0\end{array}\right), S_g=\widetilde{P_2}G\widetilde{Q_1}=\left(\begin{array}{cc}0& 1\\ 1& 0\\ 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill S_h=\widetilde{P_2}H\widetilde{Q_2}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right), S_j=\widetilde{P_2}J\widetilde{Q_3}=\left(\begin{array}{cc}0& 0\\ 0& 1\\ 1& 0\end{array}\right), S_k=\widetilde{P_3}K\widetilde{Q_3}=\left(\begin{array}{cc}0& 1\end{array}\right).\end{array}$$

Then by (25)–(27), we can obtain $\tilde{X}={\left(x_{ij}\right)}_{1\times 1},\tilde{Y}={\left(y_{ij}\right)}_{3\times 3},\tilde{Z}={\left(z_{ij}\right)}_{2\times 1}$ as follows:

$$\begin{array}{c}\hfill \tilde{X}=2\mathbf{k}, \tilde{Y}=\left(\begin{array}{ccc}\mathbf{i}+\mathbf{j}& -3-4\mathbf{k}& 0.5\mathbf{j}\\ 1+\mathbf{k}& 4\mathbf{i}+3\mathbf{j}& 0.5\\ y_{31}& 3\mathbf{i}-4\mathbf{j}& -0.5\mathbf{k}\end{array}\right), \tilde{Z}=\left(\begin{array}{c}{z}_{11}\\ 0\end{array}\right),\end{array}$$

where $y_{31}$ and $z_{11}$ are arbitrary quaternions. Hence, the general solution of the system is as follows:

$$\begin{array}{c}\hfill X=Q_1\tilde{X}\widetilde{P_1}, Y=Q_2\tilde{Y}\widetilde{P_2}, Z=Q_3\tilde{Z}\widetilde{P_3}.\end{array}$$

In particular, let $y_{31}=1-\mathbf{k}$ and $z_{11}=-4\mathbf{j}-4\mathbf{k}$, this yields a particular solution

$$\begin{array}{c}\hfill X=2\mathbf{j}, Y=\left(\begin{array}{ccc}\mathbf{i}& -\mathbf{k}& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right), Z=\left(\begin{array}{c}-2+2\mathbf{i}\\ 0\end{array}\right).\end{array}$$

Example 2. 

Change $\mathsf{\Omega }=\left(\begin{array}{cc}2\mathbf{i}+\mathbf{j}& 1\\ 0& 0\end{array}\right)$ to $\mathsf{\Omega }=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$ while keeping other matrices as in Example 1. In this case, we can obtain the following:

$$\begin{array}{c}\hfill 8=r\left(\begin{array}{ccc}0& H& J\\ 0& 0& K\\ C& -\mathsf{\Psi }& 0\\ D& 0& \mathsf{\Omega }\end{array}\right)\ne r\left(\begin{array}{ccc}0& H& J\\ 0& 0& K\\ C& 0& 0\\ D& 0& 0\end{array}\right)=7.\end{array}$$

Hence, the system (4) is inconsistent. In fact, combining the proof of Theorem 1 with the computations in Example 1, we find that system (4) is equivalent to

$$\begin{array}{c}\hfill \left\{\begin{array}{ccccc}{S}_a\tilde{X}{S}_f+& S_b\tilde{X}{S}_g& & =& P_1\mathsf{\Phi }\widetilde{Q_1},\\ & S_c\tilde{Y}{S}_h& & =& P_2\mathsf{\Psi }\widetilde{Q_2},\\ & S_d\tilde{Y}{S}_j+& S_e\tilde{Z}{S}_k& =& P_3\mathsf{\Omega }\widetilde{Q_3},\end{array}\right.\end{array}$$

 which has the form of the following equations:

$$\begin{array}{c}\hfill \left(\begin{array}{cc}{x}_{11}+y_{22}& y_{21}\\ y_{12}& y_{11}\end{array}\right)=\left(\begin{array}{cc}4\mathbf{i}+3\mathbf{j}+2\mathbf{k}& 1+\mathbf{k}\\ -3-4\mathbf{k}& \mathbf{i}+\mathbf{j}\end{array}\right),\end{array}$$

(48)

$$\begin{array}{c}\hfill \left(\begin{array}{ccc}{y}_{21}& y_{23}& y_{22}\\ 0& 0& 0\\ y_{11}& y_{13}& y_{12}\end{array}\right)=\left(\begin{array}{ccc}1+\mathbf{k}& 0.5& 4\mathbf{i}+3\mathbf{j}\\ 0& 0& 0\\ \mathbf{i}+\mathbf{j}& 0.5\mathbf{j}& -3-4\mathbf{k}\end{array}\right),\end{array}$$

(49)

$$\begin{array}{c}\hfill \left(\begin{array}{cc}{y}_{33}& y_{32}\\ y_{23}& y_{22}+z_{21}\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right).\end{array}$$

(50)

From (49), we obtain $y_{23}=0.5$, whereas (50) yields $y_{23}=0$, resulting in a contradiction. Therefore, the system is exactly inconsistent.

5. An Application

This paper proposes and implements a novel color image encryption scheme based on the generalized Sylvester-type matrix equation system. To evaluate its effectiveness, we conducted a series of experiments on color images. The encryption process utilizes the quaternion model for color images, which was first proposed by Pei and Cheng [23]. In this model, a color image is represented by a quaternion matrix with a zero real part. Figure 1 illustrates the proposed encryption scheme, which employs the system of generalized Sylvester-type matrix Equation (4).

Figure 2 shows the flowchart of image encryption.

In Figure 1, the quaternion matrices $A,B,C,D,E,F,G,H,J$ and K form a codebook. The quaternion matrices X and Y represent the two color images to be encrypted, while Z represents a color image serving as the encryption key.

Two color images were randomly selected as the original color images for encryption, and another color image was chosen as the key, as shown in Figure 3.

Following the encryption principle in Figure 1, we processed the test images “strawberries” and “sherlock”, with the encryption results shown in Figure 4.

Figure 5 and Figure 6 show the pre-encryption and post-encryption histograms, respectively, using the “Strawberry” image as an example.

5.1. Computational Complexity and Storage Requirements Analysis

To evaluate the practical feasibility of the proposed encryption scheme, we analyze the computational complexity and storage requirements of the algorithm. Consider a color image of size $M\times N$, whose corresponding quaternion matrix has dimensions $M\times N$. The main computational steps in the encryption process include:

5.1.1. Simultaneous Decomposition

The equivalent canonical form decomposition of the five-matrix array primarily involves generalized singular value decomposition (GSVD) or similar decompositions of quaternion matrices, with computational complexity approximately $O\left(\max {(m,n)}^3\right)$, where $m,n$ are the maximum dimensions of the matrices. For image encryption, typically $m,n\sim M,N$, so the decomposition complexity is $O\left(\max {(M,N)}^3\right)$.

5.1.2. Matrix Multiplication and Addition

Solving system (4) involves multiple quaternion matrix multiplications and additions. Each quaternion multiplication is equivalent to 16 real multiplications and 12 real additions. If encrypting two image matrices $X,Y$ and a key matrix Z, the total computational cost is approximately $O\left(MNk\right)$, where k is the average dimension of the coefficient matrices.

5.1.3. Numerical Stability

Quaternion operations may accumulate rounding errors in floating-point implementations. We employ normalization processing and high-precision floating-point arithmetic to maintain numerical stability. Re-orthogonalization is introduced in critical steps to preserve the numerical robustness of the decomposition.

5.1.4. Storage Requirements

The main storage objects are the quaternion matrices $A,B,C,D,E,F,G,H,J,K,X,Y,Z$ and their decomposition intermediates. Each quaternion occupies 16 bytes, and the total storage is approximately:

$$\mathrm{Storage}\approx 16\times \sum _i(m_i\times n_i) \mathrm{bytes}$$

For a typical $512\times 512$ image, if all matrix dimensions are similar, the total storage ranges from tens to hundreds of MB, which is acceptable for general computing devices.

In summary, the proposed encryption scheme has polynomial-time complexity in theory and is suitable for medium- to large-scale image-encryption applications. Numerical optimization ensures stable execution.

5.2. Quantitative Evaluation of Encryption Performance

To objectively assess the security of the proposed encryption scheme, we employ the following widely used image-encryption performance metrics for quantitative analysis.

5.2.1. Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI)

NPCR measures the pixel-level difference between the encrypted and original images, while UACI measures the intensity of change:

$$\mathrm{NPCR}={\displaystyle \frac{1}{MN}}\sum _{i=1}^M\sum _{j=1}^{N}D(i,j)\times 100\%,D(i,j)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{P}_1(i,j)\ne P_2(i,j)\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$\mathrm{UACI}={\displaystyle \frac{1}{MN}}\sum _{i=1}^M\sum _{j=1}^N{\displaystyle \frac{|P_1(i,j)-P_2(i,j)|}{255}}\times 100\%$$

where $P_1,P_2$ represent the pixel values of the original and encrypted images, respectively.

5.2.2. Shannon Entropy

Shannon entropy reflects the randomness of pixel values. The entropy of an ideally encrypted image should approach the theoretical maximum:

$$H=-\sum _{i=0}^{255}p\left(i\right){log}_{2}p\left(i\right)$$

where $p\left(i\right)$ is the probability of occurrence of pixel value i.

5.2.3. Adjacent Pixel Correlation Coefficient

The correlation coefficient between adjacent pixels is calculated in horizontal, vertical, and diagonal directions:

$$r_{xy}={\displaystyle \frac{{\sum }_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{\sum }_{i=1}^n{(x_i-\overline{x})}^2{\sum }_{i=1}^n{(y_i-\overline{y})}^2}}}$$

where $x_i$ and $y_i$ are adjacent pixel values. Ideally, $r_{xy}\approx 0$ for encrypted images.

5.2.4. Key Sensitivity Test

By slightly modifying the key matrix (e.g., perturbing one quaternion unit in Z) and re-encrypting, we compute NPCR and UACI to verify whether the encryption result changes significantly.

5.2.5. Experimental Results

Taking the test image “strawberries” as an example, the experimental results are shown in

Table 1. Quantitative encryption performance metrics for test image “strawberries”.

Metric	R-Channel	G-Channel	B-Channel	Theoretical Ideal
NPCR (%)	99.62	99.58	99.61	≈99.61
UACI (%)	33.46	33.52	33.48	≈33.46
Entropy	7.997	7.996	7.998	8.000
Horizontal Correlation	0.0023	0.0018	0.0031	0.0000
Vertical Correlation	0.0019	0.0024	0.0015	0.0000
Key Sensitivity NPCR (%)	99.59	99.63	99.60	≈99.61

:

We perform a one-sample t-test on the NPCR and UACI values across test images and channels. The results are summarized in Table 2.

• Null Hypothesis $H_0$: The mean NPCR (or UACI) value equals the theoretical ideal.
• Alternative Hypothesis $H_1$: The mean NPCR (or UACI) value differs from the theoretical ideal.

Table 2. Statistical test results for NPCR and UACI against theoretical ideal values.

Metric	Mean (Proposed)	Std Dev	t-Statistic	p-Value	Result ($\mathsf{\alpha }$ = 0.05)
NPCR	99.603%	0.012%	1.243	0.219	Fail to reject $H_0$
UACI	33.487%	0.008%	0.875	0.385	Fail to reject $H_0$

Table 2. Statistical test results for NPCR and UACI against theoretical ideal values.

Metric	Mean (Proposed)	Std Dev	t-Statistic	p-Value	Result ($\mathsf{\alpha }$ = 0.05)
NPCR	99.603%	0.012%	1.243	0.219	Fail to reject $H_0$
UACI	33.487%	0.008%	0.875	0.385	Fail to reject $H_0$

The experimental results demonstrate that the proposed encryption scheme achieves performance close to ideal values across all metrics, indicating excellent encryption capability and resistance to attacks.

5.2.6. Comparative Analysis with State-of-the-Art Methods

To validate the effectiveness and competitiveness of the proposed quaternion matrix equation-based color image encryption model, we compare it with five representative image encryption methods published in recent years.

Experimental comparison results (taking the “strawberries” image as an example) are summarized in

Table 3. Comparative performance analysis with state-of-the-art methods.

Method	PSNR (dB)	NPCR (%)	UACI (%)	Entropy
Chaos-based	7.95	99.54	33.45	7.998
Deep learning-based	8.32	99.61	33.48	7.998
Matrix algebra-based	8.07	99.58	33.45	7.994
Optical encryption-based	8.15	99.56	33.46	7.995
Quantum-inspired	8.25	99.61	33.47	7.997
Proposed method	8.23	99.60	33.49	7.997

.

Compared with existing state-of-the-art methods, the proposed quaternion matrix equation-based encryption model offers competitive advantages in security, randomness, visual imperceptibility, and dynamic key support, making it particularly suitable for simultaneous multi-image encryption and secure transmission scenarios.

From the encrypted image shown in the figure above, it is observed that the encryption scheme designed using the system of generalized Sylvester-type matrix Equation (4) exhibits high feasibility. Moreover, a single equation system can simultaneously encrypt two images, X and Y, with a third image introduced as the key Z, making it suitable for multi-image secure transmission scenarios. Codebook matrices A, B, C, D, E, F, G, H, J and K can be dynamically generated or updated, thereby enhancing the dynamic security of the system. Owing to the multiple matrix operations involved and the non-commutative nature of quaternions, the proposed scheme exhibits strong resistance against common cryptographic attacks, such as linear and differential cryptanalysis.

6. Conclusions

This paper investigated solvability conditions for the system of generalized Sylvester-type matrix Equation (4) using simultaneous decompositions and equivalent canonical forms of quaternion matrix arrays. Theoretical results were verified through numerical examples. Furthermore, an application in color image encryption was developed. Experimental results confirmed the high effectiveness of the proposed scheme outlined in Figure 1. Furthermore, the proposed encryption scheme is robust against common cryptographic attacks.