The Solvability of a System of Quaternion Matrix Equations Involving ϕ-Skew-Hermicity

Abstract

Let $\mathbb{H}$ be the real quaternion algebra and ${\mathbb{H}}^{m\times n}$ denote the set of all $m\times n$ matrices over $\mathbb{H}$. For $A\in {\mathbb{H}}^{m\times n},$ we denote by $A_{\varphi }$ the $n\times m$ matrix obtained by applying $\varphi$ entrywise to the transposed matrix $A^T,$ where $\varphi$ is a non-standard involution of $\mathbb{H}$. $A\in {\mathbb{H}}^{n\times n}$ is said to be $\varphi$-skew-Hermicity if $A=-A_{\varphi }$. In this paper, we provide some necessary and sufficient conditions for the existence of a $\varphi$-skew-Hermitian solution to the system of quaternion matrix equations with four unknowns $A_i{X}_i{\left(A_i\right)}_{\varphi }+B_i{X}_{i+1}{\left(B_i\right)}_{\varphi }=C_i,(i=1,2,3),A_4{X}_4{\left(A_4\right)}_{\varphi }=C_4$.

1. Introduction

Let $\mathbb{R}$ denote the field of real numbers, $\mathbb{H}$ be a four-dimensional vector space over $\mathbb{R}$ with an ordered basis $\{1,\mathbf{i},\mathbf{j},\mathbf{k}\}$. A real quaternion, simply called quaternion, is a vector $x=a_0+a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}\in \mathbb{H}$ with real coefficients $a_0,a_1,a_2,a_3$. Moreover, $\mathbf{i},\mathbf{j},\mathbf{k}$ satisfies

$${\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=-1,$$

$$\mathbf{i}\mathbf{j}=-\mathbf{j}\mathbf{i}=\mathbf{k},\mathbf{j}\mathbf{k}=-\mathbf{k}\mathbf{j}=\mathbf{i},\mathbf{k}\mathbf{i}=-\mathbf{i}\mathbf{k}=\mathbf{j}.$$

Let $\mathbb{R}$ and ${\mathbb{H}}^{m\times n}$ stand, respectively, for the real number field and the set of all $m\times n$ matrices over the real quaternion algebra

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

The definitions of $\varphi$-skew-Hermitian quaternion matrices were first introduced by Rodman (Definition 3.6.1 in [1]). For $A\in {\mathbb{H}}^{m\times n},$ we denote by $A_{\varphi }$ the $n\times m$ matrix obtained by applying $\varphi$ entrywise to the transposed matrix $A^T,$ where $\varphi$ is a non-standard involution of $\mathbb{H}$ (see Definition 1). $A\in {\mathbb{H}}^{n\times n}$ is said to be $\varphi$-skew-Hermicity if $A=-A_{\varphi }$.

The decompositions of the quaternion matrices have applications in many fields, such as color image processing(e.g., [2,3]), quantum mechanics [4], signal processing [5], and so on. Research on quaternion matrix theories (e.g., [6,7,8,9,10,11,12,13,14,15,16,17,18]) and equations (e.g., [13,19,20,21,22,23,24,25,26,27]) is ongoing.

The quaternion matrix equation involving Hermicity is one of the active research topics in the matrix field and its applications. Wang and Zhang [28] provided necessary and sufficient conditions for the existence and expression of the Re-nonnegative definite solution to the system

$$AX{A}^{*}+BY{B}^{*}=C$$

over $\mathbb{H}$ by using the decomposition of pairwise matrices, where * stands for conjugate transpose. Wang and Jiang [29] further studied the extreme ranks of the (skew-)Hermitian solutions to the quaternion matrix equation. He [30] investigated the system of coupled real quaternion matrix equations involving $\eta$-Hermicity

$$A_i{X}_i{A}_i^{\eta *}+B_i{X}_{i+1}{B}_i^{\eta *}=C_i,(i=1,2,3),$$

where $A_i\in {\mathbb{H}}^{p_i\times t_i},B_i\in {\mathbb{H}}^{p_i\times t_{i+1}},C_i\in {\mathbb{H}}^{p_i\times p_i}$, and $C_i$ are $\eta$-Hermitian matrices. He gave the solvability conditions, general solutions, and the rank bounds of the general $\eta$-Hermitian solutions. Some researchers have considered the $\varphi$-skew-Hermitian solution to some quaternion matrix equations. For example, He [31] derived some necessary and sufficient conditions for the existence of a $\varphi$-skew-Hermitian solution to the following system of quaternion matrix equations involving $\varphi$-skew-Hermicity

$$\begin{array}{c}\hfill \left\{\begin{array}{c}BX{B}_{\varphi }+CY{C}_{\varphi }=A,\\ EZ{E}_{\varphi }+DY{D}_{\varphi }=F,\end{array}\right.X=-X_{\varphi },Y=-Y_{\varphi },Z=-Z_{\varphi },\end{array}$$

where A, B, C, D, E, F are the given quaternion matrices.

To our knowledge, there is little information on the system of quaternion matrix equations involving $\varphi$-skew-Hermicity with four unknowns

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{A}_1{X}_1{\left(A_1\right)}_{\varphi }+B_1{X}_2{\left(B_1\right)}_{\varphi }=C_1,\\ A_2{X}_2{\left(A_2\right)}_{\varphi }+B_2{X}_3{\left(B_2\right)}_{\varphi }=C_2,\\ A_3{X}_3{\left(A_3\right)}_{\varphi }+B_3{X}_4{\left(B_3\right)}_{\varphi }=C_3,\\ A_4{X}_4{\left(A_4\right)}_{\varphi }=C_4,\end{array}\right.X_i=-{\left(X_i\right)}_{\varphi },\end{array}$$

(1)

where $A_i\in {\mathbb{H}}^{p_i\times t_i}$, $B_i\in {\mathbb{H}}^{p_i\times t_{i+1}}$, $C_i\in {\mathbb{H}}^{p_i\times p_i}$, and $C_i$ are $\varphi$-skew-Hermitian matrices. Using the simultaneous decomposition of a set of seven real quaternion matrices

$$\begin{array}{c}\hfill \left(\begin{array}{cccc}{A}_1& B_1& & \\ & A_2& B_2& \\ & & A_3& B_3\\ & & & A_4\end{array}\right),\end{array}$$

(2)

we provide some necessary and sufficient conditions for the existence of a $\varphi$-skew-Hermitian solution to the system (1).

The remainder of this paper is organized as follows. In Section 2, we review the definitions of the non-standard involution $\varphi$ and the $\varphi$-skew-Hermitian quaternion matrix; we also provide a simultaneous decomposition for a set of eleven real quaternion matrices involving $\varphi$-skew-Hermicity and present a canonical form of the system of the quaternion matrix, Equation (1). In Section 3, we provide some necessary and sufficient conditions for the existence of a $\varphi$-skew-Hermitian solution to the system (1).

2. A Canonical Form of the System of the Quaternion Matrix Equation

In this section, we investigate the structure of a simultaneous decomposition for the matrix array (2) and provide a canonical form of the system of the quaternion matrix Equations (1). First, we review the definitions of non-standard involution $\varphi$ and $\varphi$-skew-Hermitian matrix.

Definition 1. 

(Non-standard involution [1]). Let ϕ be an anti-endomorphism of $\mathbb{H}$. Assume that ϕ does not map $\mathbb{H}$ into zero. Then, ϕ is one-to-one and onto $\mathbb{H}$; thus, ϕ is an anti-automorphism. Moreover, ϕ is real linear and can be represented as a $4\times 4$ real matrix with respect to the basis $\{1,\mathbf{i},\mathbf{j},\mathbf{k}\}$. Then, ϕ is a non-standard involution if and only if

$$\varphi =\left(\begin{array}{cc}1& 0\\ 0& T\end{array}\right),$$

where T is a $3\times 3$ real orthogonal symmetric matrix with eigenvalues $1,1,-1$.

Definition 2. 

($\mathbf{\varphi }$-skew-Hermitian [1]). A ∈${\mathbb{H}}^{n\times n}$ is said to be ϕ-skew-Hermitian if $A=-{\left(A\right)}_{\varphi }$, where ϕ is a non-standard involution.

The following Theorem presents the equivalence canonical form of the set of seven real quaternion matrices (2).

Using the results of [31,32], we can obtain the following result.

Lemma 1. 

Consider a set of seven matrices (2); there exists a unitary matrix $\widehat{T_1}\in {\mathbb{H}}^{t_1\times t_1},$ nonsingular matrices $\widehat{P_i}\in {\mathbb{H}}^{p_i\times p_i},(i=1,2,3),\widehat{T_2}\in {\mathbb{H}}^{t_2\times t_2},\widehat{T_3}\in {\mathbb{H}}^{t_3\times t_3},\widehat{T_4}\in {\mathbb{H}}^{t_4\times t_4},\widehat{P_4}\in {\mathbb{H}}^{p_4\times p_4}$ such that

$$\begin{array}{c}\hfill \widehat{P_i}{A}_i\widehat{T_i}=S_{a_i},\widehat{P_i}{B}_i\widehat{T_{i+1}}=S_{b_i},\widehat{P_4}{A}_4\widehat{T_4}=S_{a_4}\end{array}$$

where

$$\begin{array}{cc}\hfill & \left(\begin{array}{cccc}{S}_{a_1}& S_{b_1}& & \\ & S_{a_2}& S_{b_2}& \\ & & S_{a_3}& S_{b_3}\\ & & & S_{a_4}\end{array}\right)\hfill \\ & =\left(\begin{array}{ccc}\begin{array}{c}\end{array}& \begin{array}{c}\begin{array}{cccccccc}I& 0& 0& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& 0& 0\\ 0& 0& 0& I& 0& 0& 0& 0\\ 0& 0& 0& 0& I& 0& 0& 0\\ 0& 0& 0& 0& 0& I& 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\end{array}\end{array}& \begin{array}{c}\begin{array}{cccccccccccccccccc}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& 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\begin{array}{c}\begin{array}{cccccccccccccccccccc}I& 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& 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& I& 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& I& 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& I& 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& I& 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\end{array}\end{array}& \begin{array}{c}\begin{array}{cccccccccccccc}I& 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& I& 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& I& 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& I& 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& I& 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& I& 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\end{array}\end{array}\\ \begin{array}{c}\end{array}& & & & \begin{array}{c}\begin{array}{cccccccccccccc}I& 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& I& 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& I& 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& I& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}\end{array}\right).\hfill \end{array}$$

(3)

It follows from Lemma 1 that the system (1) becomes

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\widehat{P_1}{A}_1\widehat{T_1}{\widehat{T_1}}^{-1}{X}_1{\left(\widehat{T_1}\right)}_{\varphi }^{-1}{\left(\widehat{T_1}\right)}_{\varphi }{\left(A_1\right)}_{\varphi }{\left(\widehat{P_1}\right)}_{\varphi }+\widehat{P_1}{B}_1\widehat{T_2}{\widehat{T_2}}^{-1}{X}_2{\left(\widehat{T_2}\right)}_{\varphi }^{-1}{\left(\widehat{T_2}\right)}_{\varphi }{\left(B_1\right)}_{\varphi }{\left(\widehat{P_1}\right)}_{\varphi }=\widehat{P_1}{C}_1{\left(\widehat{P_1}\right)}_{\varphi },\\ \widehat{P_2}{A}_2\widehat{T_2}{\widehat{T_2}}^{-1}{X}_2{\left(\widehat{T_2}\right)}_{\varphi }^{-1}{\left(\widehat{T_2}\right)}_{\varphi }{\left(A_2\right)}_{\varphi }{\left(\widehat{P_2}\right)}_{\varphi }+\widehat{P_2}{B}_2\widehat{T_3}{\widehat{T_3}}^{-1}{X}_3{\left(\widehat{T_3}\right)}_{\varphi }^{-1}{\left(\widehat{T_3}\right)}_{\varphi }{\left(B_2\right)}_{\varphi }{\left(\widehat{P_2}\right)}_{\varphi }=\widehat{P_2}{C}_2{\left(\widehat{P_2}\right)}_{\varphi },\\ \widehat{P_3}{A}_3\widehat{T_3}{\widehat{T_3}}^{-1}{X}_3{\left(\widehat{T_3}\right)}_{\varphi }^{-1}{\left(\widehat{T_3}\right)}_{\varphi }{\left(A_3\right)}_{\varphi }{\left(\widehat{P_3}\right)}_{\varphi }+\widehat{P_3}{B}_3\widehat{T_4}{\widehat{T_4}}^{-1}{X}_4{\left(\widehat{T_4}\right)}_{\varphi }^{-1}{\left(\widehat{T_4}\right)}_{\varphi }{\left(B_3\right)}_{\varphi }{\left(\widehat{P_3}\right)}_{\varphi }=\widehat{P_3}{C}_3{\left(\widehat{P_3}\right)}_{\varphi },\\ \widehat{P_4}{A}_4\widehat{T_4}{\widehat{T_4}}^{-1}{X}_4{\left(\widehat{T_4}\right)}_{\varphi }^{-1}{\left(\widehat{T_4}\right)}_{\varphi }{\left(A_4\right)}_{\varphi }{\left(\widehat{P_4}\right)}_{\varphi }=\widehat{P_4}{C}_4{\left(\widehat{P_4}\right)}_{\varphi },\end{array}\right.\end{array}$$

where $X_i=-{\left(X_i\right)}_{\varphi }$.

Put

$$\widehat{X_1}={\widehat{T_1}}^{-1}{X}_1{\left(\widehat{T_1}\right)}_{\varphi }^{-1},\widehat{X_2}={\widehat{T_2}}^{-1}{X}_2{\left(\widehat{T_2}\right)}_{\varphi }^{-1},\widehat{X_3}={\widehat{T_3}}^{-1}{X}_3{\left(\widehat{T_3}\right)}_{\varphi }^{-1},\widehat{X_4}={\widehat{T_4}}^{-1}{X}_4{\left(\widehat{T_4}\right)}_{\varphi }^{-1},$$

$$D_{ij}^{\left(1\right)}=\widehat{P_1}{C}_1{\left(\widehat{P_1}\right)}_{\varphi },D_{ij}^{\left(2\right)}=\widehat{P_2}{C}_2{\left(\widehat{P_2}\right)}_{\varphi },D_{ij}^{\left(3\right)}=\widehat{P_3}{C}_3{\left(\widehat{P_3}\right)}_{\varphi },D_{ij}^{\left(4\right)}=\widehat{P_4}{C}_4{\left(\widehat{P_4}\right)}_{\varphi }.$$

According to Lemma 1, the system (1) is equivalent to the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{S}_{a_1}\widehat{X_1}{\left(S_{a_1}\right)}_{\varphi }+S_{b_1}\widehat{X_2}{\left(S_{b_1}\right)}_{\varphi }=D_{ij}^{\left(1\right)},\\ S_{a_2}\widehat{X_2}{\left(S_{a_2}\right)}_{\varphi }+S_{b_2}\widehat{X_3}{\left(S_{b_2}\right)}_{\varphi }=D_{ij}^{\left(2\right)},\\ S_{a_3}\widehat{X_3}{\left(S_{a_3}\right)}_{\varphi }+S_{b_3}\widehat{X_4}{\left(S_{b_3}\right)}_{\varphi }=D_{ij}^{\left(3\right)},\\ S_{a_4}\widehat{X_4}{\left(S_{a_4}\right)}_{\varphi }=D_{ij}^{\left(4\right)},\end{array}\right.\end{array}$$

(4)

where $X_i=-{\left(X_i\right)}_{\varphi }$. In the next section, we will consider the system (4).

3. Solvability Conditions for the Quaternion Matrix Equation to Possess a $\mathbf{\varphi }$-Skew-Hermitian Solution

In this section, we provide some necessary and sufficient conditions for the existence of a $\varphi$-skew-Hermitian solution to the system (1). In order to solve the system of quaternion matrix Equation (1), we need to solve the system of quaternion matrix Equation (4).

First, let matrices $\widehat{X_1}$, $\widehat{X_2}$, $\widehat{X_3}$, $\widehat{X_4}$ have the following forms:

$$\begin{array}{c}\hfill \widehat{X_1}=-{\left(\widehat{X_1}\right)}_{\varphi }=\left(\begin{array}{cccc}{X}_{11}^{\left(1\right)}& X_{12}^{\left(1\right)}& \cdots & X_{18}^{\left(1\right)}\\ -{\left(X_{12}^{\left(1\right)}\right)}_{\varphi }& X_{22}^{\left(1\right)}& \cdots & X_{28}^{\left(1\right)}\\ ⋮& ⋮& \cdots & ⋮\\ -{\left(X_{18}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{28}^{\left(1\right)}\right)}_{\varphi }& \cdots & X_{88}^{\left(1\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill \widehat{X_2}=-{\left(\widehat{X_2}\right)}_{\varphi }=\left(\begin{array}{cccc}{X}_{11}^{\left(2\right)}& X_{12}^{\left(2\right)}& \cdots & X_{1,18}^{\left(2\right)}\\ -{\left(X_{12}^{\left(2\right)}\right)}_{\varphi }& X_{22}^{\left(2\right)}& \cdots & X_{2,18}^{\left(2\right)}\\ ⋮& ⋮& \cdots & ⋮\\ -{\left(X_{1,18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,18}^{\left(2\right)}\right)}_{\varphi }& \cdots & X_{18,18}^{\left(2\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill \widehat{X_3}=-{\left(\widehat{X_3}\right)}_{\varphi }=\left(\begin{array}{cccc}{X}_{11}^{\left(3\right)}& X_{12}^{\left(3\right)}& \cdots & X_{1,20}^{\left(3\right)}\\ -{\left(X_{12}^{\left(3\right)}\right)}_{\varphi }& X_{22}^{\left(3\right)}& \cdots & X_{2,20}^{\left(3\right)}\\ ⋮& ⋮& \cdots & ⋮\\ -{\left(X_{1,20}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,20}^{\left(3\right)}\right)}_{\varphi }& \cdots & X_{20,20}^{\left(3\right)}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill \widehat{X_4}=-{\left(\widehat{X_4}\right)}_{\varphi }=\left(\begin{array}{cccc}{X}_{11}^{\left(4\right)}& X_{12}^{\left(4\right)}& \cdots & X_{1,14}^{\left(4\right)}\\ -{\left(X_{12}^{\left(4\right)}\right)}_{\varphi }& X_{22}^{\left(4\right)}& \cdots & X_{2,14}^{\left(4\right)}\\ ⋮& ⋮& \cdots & ⋮\\ -{\left(X_{1,14}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{2,14}^{\left(4\right)}\right)}_{\varphi }& \cdots & X_{14,14}^{\left(4\right)}\end{array}\right).\end{array}$$

Then, substituting $\widehat{X_1}$ and $\widehat{X_2}$ into the first equation in (4) yields

$$\begin{array}{c}\hfill {\left(D_{ij}^{\left(1\right)}\right)}_{14\times 14}=(D_1^{\left(1\right)},D_2^{\left(1\right)}),\end{array}$$

(5)

where

$$\begin{array}{c}\hfill D_1^{\left(1\right)}=\begin{array}{c}\hfill \left(\begin{array}{ccccc}{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_{12}^{\left(1\right)}+X_{12}^{\left(2\right)})}_{\varphi }& 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_{13}^{\left(1\right)}+X_{13}^{\left(2\right)})}_{\varphi }& -{(X_{23}^{\left(1\right)}+X_{23}^{\left(2\right)})}_{\varphi }& 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_{14}^{\left(1\right)}+X_{14}^{\left(2\right)})}_{\varphi }& -{(X_{24}^{\left(1\right)}+X_{24}^{\left(2\right)})}_{\varphi }& -{(X_{34}^{\left(1\right)}+X_{34}^{\left(2\right)})}_{\varphi }& X_{44}^{\left(1\right)}+X_{44}^{\left(2\right)}& X_{45}^{\left(1\right)}+X_{45}^{\left(2\right)}\\ -{(X_{15}^{\left(1\right)}+X_{15}^{\left(2\right)})}_{\varphi }& -{(X_{25}^{\left(1\right)}+X_{25}^{\left(2\right)})}_{\varphi }& -{(X_{35}^{\left(1\right)}+X_{35}^{\left(2\right)})}_{\varphi }& -{(X_{45}^{\left(1\right)}+X_{45}^{\left(2\right)})}_{\varphi }& X_{55}^{\left(1\right)}+X_{55}^{\left(2\right)}\\ -{(X_{16}^{\left(1\right)}+X_{16}^{\left(2\right)})}_{\varphi }& -{(X_{26}^{\left(1\right)}+X_{26}^{\left(2\right)})}_{\varphi }& -{(X_{36}^{\left(1\right)}+X_{36}^{\left(2\right)})}_{\varphi }& -{(X_{46}^{\left(1\right)}+X_{46}^{\left(2\right)})}_{\varphi }& -{(X_{56}^{\left(1\right)}+X_{56}^{\left(2\right)})}_{\varphi }\\ -{\left(X_{17}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{27}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{37}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{47}^{\left(1\right)}\right)}_{\varphi }& -{\left(X_{57}^{\left(1\right)}\right)}_{\varphi }\\ -{\left(X_{17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{27}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{37}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{47}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{57}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{18}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{28}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{38}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{48}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{58}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{19}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{29}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{39}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{49}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{59}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{1,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{3,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{4,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{5,10}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{1,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{3,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{4,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{5,11}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{1,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{3,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{4,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{5,12}^{\left(2\right)}\right)}_{\varphi }\\ 0& 0& 0& 0& 0\end{array}\right),\end{array}\end{array}$$

$$\begin{array}{c}\hfill D_2^{\left(1\right)}=\begin{array}{c}\hfill \left(\begin{array}{ccccccccc}{X}_{16}^{\left(1\right)}+X_{16}^{\left(2\right)}& X_{17}^{\left(1\right)}& X_{17}^{\left(2\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)}& 0\\ X_{26}^{\left(1\right)}+X_{26}^{\left(2\right)}& X_{27}^{\left(1\right)}& X_{27}^{\left(2\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)}& 0\\ X_{36}^{\left(1\right)}+X_{36}^{\left(2\right)}& X_{37}^{\left(1\right)}& X_{37}^{\left(2\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)}& 0\\ X_{46}^{\left(1\right)}+X_{46}^{\left(2\right)}& X_{47}^{\left(1\right)}& X_{47}^{\left(2\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)}& 0\\ X_{56}^{\left(1\right)}+X_{56}^{\left(2\right)}& X_{57}^{\left(1\right)}& X_{57}^{\left(2\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)}& 0\\ X_{66}^{\left(1\right)}+X_{66}^{\left(2\right)}& X_{67}^{\left(1\right)}& X_{67}^{\left(2\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)}& 0\\ -{\left(X_{67}^{\left(1\right)}\right)}_{\varphi }& X_{77}^{\left(1\right)}& 0& 0& 0& 0& 0& 0& 0\\ -{\left(X_{67}^{\left(2\right)}\right)}_{\varphi }& 0& X_{77}^{\left(2\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)}& 0\\ -{\left(X_{68}^{\left(2\right)}\right)}_{\varphi }& 0& -{\left(X_{78}^{\left(2\right)}\right)}_{\varphi }& 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)}& 0\\ -{\left(X_{69}^{\left(2\right)}\right)}_{\varphi }& 0& -{\left(X_{79}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{89}^{\left(2\right)}\right)}_{\varphi }& X_{9,9}^{\left(2\right)}& X_{9,10}^{\left(2\right)}& X_{9,11}^{\left(2\right)}& X_{9,12}^{\left(2\right)}& 0\\ -{\left(X_{6,10}^{\left(2\right)}\right)}_{\varphi }& 0& -{\left(X_{7,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{8,10}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{9,10}^{\left(2\right)}\right)}_{\varphi }& X_{10,10}^{\left(2\right)}& X_{10,11}^{\left(2\right)}& X_{10,12}^{\left(2\right)}& 0\\ -{\left(X_{6,11}^{\left(2\right)}\right)}_{\varphi }& 0& -{\left(X_{7,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{8,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{9,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{10,11}^{\left(2\right)}\right)}_{\varphi }& X_{11,11}^{\left(2\right)}& X_{11,12}^{\left(2\right)}& 0\\ -{\left(X_{6,12}^{\left(2\right)}\right)}_{\varphi }& 0& -{\left(X_{7,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{8,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{9,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{10,12}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{11,12}^{\left(2\right)}\right)}_{\varphi }& X_{12,12}^{\left(2\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right).\end{array}\end{array}$$

Substituting $\widehat{X_2}$ and $\widehat{X_3}$ into the second Equation in (4) yields

$$\begin{array}{c}\hfill {\left(D_{ij}^{\left(2\right)}\right)}_{20\times 20}=(D_1^{\left(2\right)},D_2^{\left(2\right)},D_3^{\left(2\right)}),\end{array}$$

(6)

where

$$\begin{array}{c}\hfill D_1^{\left(2\right)}=\begin{array}{c}\hfill \left(\begin{array}{cccccc}{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_{17}^{\left(2\right)}+X_{15}^{\left(3\right)}\\ -{(X_{12}^{\left(2\right)}+X_{12}^{\left(3\right)})}_{\varphi }& 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_{27}^{\left(2\right)}+X_{25}^{\left(3\right)}\\ -{(X_{13}^{\left(2\right)}+X_{13}^{\left(3\right)})}_{\varphi }& -{(X_{23}^{\left(2\right)}+X_{23}^{\left(3\right)})}_{\varphi }& 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_{37}^{\left(2\right)}+X_{35}^{\left(3\right)}\\ -{(X_{14}^{\left(2\right)}+X_{14}^{\left(3\right)})}_{\varphi }& -{(X_{24}^{\left(2\right)}+X_{24}^{\left(3\right)})}_{\varphi }& -{(X_{34}^{\left(2\right)}+X_{34}^{\left(3\right)})}_{\varphi }& X_{44}^{\left(2\right)}+X_{44}^{\left(3\right)}& X_{45}^{\left(2\right)}& X_{47}^{\left(2\right)}+X_{45}^{\left(3\right)}\\ -{\left(X_{15}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{25}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{35}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{45}^{\left(2\right)}\right)}_{\varphi }& X_{55}^{\left(2\right)}& X_{57}^{\left(2\right)}\\ -{(X_{17}^{\left(2\right)}+X_{15}^{\left(3\right)})}_{\varphi }& -{(X_{27}^{\left(2\right)}+X_{25}^{\left(3\right)})}_{\varphi }& -{(X_{37}^{\left(2\right)}+X_{35}^{\left(3\right)})}_{\varphi }& -{(X_{47}^{\left(2\right)}+X_{45}^{\left(3\right)})}_{\varphi }& -{\left(X_{57}^{\left(2\right)}\right)}_{\varphi }& X_{77}^{\left(2\right)}+X_{55}^{\left(3\right)}\\ -{(X_{18}^{\left(2\right)}+X_{16}^{\left(3\right)})}_{\varphi }& -{(X_{28}^{\left(2\right)}+X_{26}^{\left(3\right)})}_{\varphi }& -{(X_{38}^{\left(2\right)}+X_{36}^{\left(3\right)})}_{\varphi }& -{(X_{48}^{\left(2\right)}+X_{46}^{\left(3\right)})}_{\varphi }& -{\left(X_{58}^{\left(2\right)}\right)}_{\varphi }& -{(X_{78}^{\left(2\right)}+X_{56}^{\left(3\right)})}_{\varphi }\\ -{(X_{19}^{\left(2\right)}+X_{17}^{\left(3\right)})}_{\varphi }& -{(X_{29}^{\left(2\right)}+X_{27}^{\left(3\right)})}_{\varphi }& -{(X_{39}^{\left(2\right)}+X_{37}^{\left(3\right)})}_{\varphi }& -{(X_{49}^{\left(2\right)}+X_{47}^{\left(3\right)})}_{\varphi }& -{\left(X_{59}^{\left(2\right)}\right)}_{\varphi }& -{(X_{79}^{\left(2\right)}+X_{57}^{\left(3\right)})}_{\varphi }\\ -{(X_{1,10}^{\left(2\right)}+X_{18}^{\left(3\right)})}_{\varphi }& -{(X_{2,10}^{\left(2\right)}+X_{28}^{\left(3\right)})}_{\varphi }& -{(X_{3,10}^{\left(2\right)}+X_{38}^{\left(3\right)})}_{\varphi }& -{(X_{4,10}^{\left(2\right)}+X_{48}^{\left(3\right)})}_{\varphi }& -{\left(X_{5,10}^{\left(2\right)}\right)}_{\varphi }& -{(X_{7,10}^{\left(2\right)}+X_{58}^{\left(3\right)})}_{\varphi }\\ -{\left(X_{1,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{3,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{4,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{5,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{7,11}^{\left(2\right)}\right)}_{\varphi }\\ -{(X_{1,13}^{\left(2\right)}+X_{19}^{\left(3\right)})}_{\varphi }& -{(X_{2,13}^{\left(2\right)}+X_{29}^{\left(3\right)})}_{\varphi }& -{(X_{3,13}^{\left(2\right)}+X_{39}^{\left(3\right)})}_{\varphi }& -{(X_{4,13}^{\left(2\right)}+X_{49}^{\left(3\right)})}_{\varphi }& -{\left(X_{5,13}^{\left(2\right)}\right)}_{\varphi }& -{(X_{7,13}^{\left(2\right)}+X_{59}^{\left(3\right)})}_{\varphi }\\ -{(X_{1,14}^{\left(2\right)}+X_{1,10}^{\left(3\right)})}_{\varphi }& -{(X_{2,14}^{\left(2\right)}+X_{2,10}^{\left(3\right)})}_{\varphi }& -{(X_{3,14}^{\left(2\right)}+X_{3,10}^{\left(3\right)})}_{\varphi }& -{(X_{4,14}^{\left(2\right)}+X_{4,10}^{\left(3\right)})}_{\varphi }& -{\left(X_{5,14}^{\left(2\right)}\right)}_{\varphi }& -{(X_{7,14}^{\left(2\right)}+X_{5,10}^{\left(3\right)})}_{\varphi }\\ -{(X_{1,15}^{\left(2\right)}+X_{1,11}^{\left(3\right)})}_{\varphi }& -{(X_{2,15}^{\left(2\right)}+X_{2,11}^{\left(3\right)})}_{\varphi }& -{(X_{3,15}^{\left(2\right)}+X_{3,11}^{\left(3\right)})}_{\varphi }& -{(X_{4,15}^{\left(2\right)}+X_{4,11}^{\left(3\right)})}_{\varphi }& -{\left(X_{5,15}^{\left(2\right)}\right)}_{\varphi }& -{(X_{7,15}^{\left(2\right)}+X_{5,11}^{\left(3\right)})}_{\varphi }\\ -{(X_{1,16}^{\left(2\right)}+X_{1,12}^{\left(3\right)})}_{\varphi }& -{(X_{2,16}^{\left(2\right)}+X_{2,12}^{\left(3\right)})}_{\varphi }& -{(X_{3,16}^{\left(2\right)}+X_{3,12}^{\left(3\right)})}_{\varphi }& -{(X_{4,16}^{\left(2\right)}+X_{4,12}^{\left(3\right)})}_{\varphi }& -{\left(X_{5,16}^{\left(2\right)}\right)}_{\varphi }& -{(X_{7,16}^{\left(2\right)}+X_{5,12}^{\left(3\right)})}_{\varphi }\\ -{\left(X_{1,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{2,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{3,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{4,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{5,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{7,17}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{1,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,13}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{5,13}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,14}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{5,14}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,15}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{5,15}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{4,16}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{5,16}^{\left(3\right)}\right)}_{\varphi }\\ 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}\end{array}$$

$$\begin{array}{c}\hfill D_2^{\left(2\right)}=\left(\begin{array}{cccccc}{X}_{18}^{\left(2\right)}+X_{16}^{\left(3\right)}& X_{19}^{\left(2\right)}+X_{17}^{\left(3\right)}& X_{1,10}^{\left(2\right)}+X_{18}^{\left(3\right)}& X_{1,11}^{\left(2\right)}& X_{1,13}^{\left(2\right)}+X_{19}^{\left(3\right)}& X_{1,14}^{\left(2\right)}+X_{1,10}^{\left(3\right)}\\ X_{28}^{\left(2\right)}+X_{26}^{\left(3\right)}& X_{29}^{\left(2\right)}+X_{27}^{\left(3\right)}& X_{2,10}^{\left(2\right)}+X_{28}^{\left(3\right)}& X_{2,11}^{\left(2\right)}& X_{2,13}^{\left(2\right)}+X_{29}^{\left(3\right)}& X_{2,14}^{\left(2\right)}+X_{2,10}^{\left(3\right)}\\ X_{38}^{\left(2\right)}+X_{36}^{\left(3\right)}& X_{39}^{\left(2\right)}+X_{37}^{\left(3\right)}& X_{3,10}^{\left(2\right)}+X_{38}^{\left(3\right)}& X_{3,11}^{\left(2\right)}& X_{3,13}^{\left(2\right)}+X_{39}^{\left(3\right)}& X_{3,14}^{\left(2\right)}+X_{3,10}^{\left(3\right)}\\ X_{48}^{\left(2\right)}+X_{46}^{\left(3\right)}& X_{49}^{\left(2\right)}+X_{47}^{\left(3\right)}& X_{4,10}^{\left(2\right)}+X_{48}^{\left(3\right)}& X_{4,11}^{\left(2\right)}& X_{4,13}^{\left(2\right)}+X_{49}^{\left(3\right)}& X_{4,14}^{\left(2\right)}+X_{4,10}^{\left(3\right)}\\ X_{58}^{\left(2\right)}& X_{59}^{\left(2\right)}& X_{5,10}^{\left(2\right)}& X_{5,11}^{\left(2\right)}& X_{5,13}^{\left(2\right)}& X_{5,14}^{\left(2\right)}\\ X_{78}^{\left(2\right)}+X_{56}^{\left(3\right)}& X_{79}^{\left(2\right)}+X_{57}^{\left(3\right)}& X_{7,10}^{\left(2\right)}+X_{58}^{\left(3\right)}& X_{7,11}^{\left(2\right)}& X_{7,13}^{\left(2\right)}+X_{59}^{\left(3\right)}& X_{7,14}^{\left(2\right)}+X_{5,10}^{\left(3\right)}\\ X_{88}^{\left(2\right)}+X_{66}^{\left(3\right)}& X_{89}^{\left(2\right)}+X_{67}^{\left(3\right)}& X_{8,10}^{\left(2\right)}+X_{68}^{\left(3\right)}& X_{8,11}^{\left(2\right)}& X_{8,13}^{\left(2\right)}+X_{69}^{\left(3\right)}& X_{8,14}^{\left(2\right)}+X_{6,10}^{\left(3\right)}\\ -{(X_{89}^{\left(2\right)}+X_{67}^{\left(3\right)})}_{\varphi }& X_{99}^{\left(2\right)}+X_{77}^{\left(3\right)}& X_{9,10}^{\left(2\right)}+X_{78}^{\left(3\right)}& X_{9,11}^{\left(2\right)}& X_{9,13}^{\left(2\right)}+X_{79}^{\left(3\right)}& X_{9,14}^{\left(2\right)}+X_{7,10}^{\left(3\right)}\\ -{(X_{8,10}^{\left(2\right)}+X_{68}^{\left(3\right)})}_{\varphi }& -{(X_{9,10}^{\left(2\right)}+X_{78}^{\left(3\right)})}_{\varphi }& X_{10,10}^{\left(2\right)}+X_{88}^{\left(3\right)}& X_{10,11}^{\left(2\right)}& X_{10,13}^{\left(2\right)}+X_{89}^{\left(3\right)}& X_{10,14}^{\left(2\right)}+X_{8,10}^{\left(3\right)}\\ -{\left(X_{8,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{9,11}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{10,11}^{\left(2\right)}\right)}_{\varphi }& X_{11,11}^{\left(2\right)}& X_{11,13}^{\left(2\right)}& X_{11,14}^{\left(2\right)}\\ -{(X_{8,13}^{\left(2\right)}+X_{69}^{\left(3\right)})}_{\varphi }& -{(X_{9,13}^{\left(2\right)}+X_{79}^{\left(3\right)})}_{\varphi }& -{(X_{10,13}^{\left(2\right)}+X_{89}^{\left(3\right)})}_{\varphi }& -{\left(X_{11,13}^{\left(2\right)}\right)}_{\varphi }& X_{13,13}^{\left(2\right)}+X_{99}^{\left(3\right)}& X_{13,14}^{\left(2\right)}+X_{9,10}^{\left(3\right)}\\ -{(X_{8,14}^{\left(2\right)}+X_{6,10}^{\left(3\right)})}_{\varphi }& -{(X_{9,14}^{\left(2\right)}+X_{7,10}^{\left(3\right)})}_{\varphi }& -{(X_{10,14}^{\left(2\right)}+X_{8,10}^{\left(3\right)})}_{\varphi }& -{\left(X_{11,14}^{\left(2\right)}\right)}_{\varphi }& -{(X_{13,14}^{\left(2\right)}+X_{9,10}^{\left(3\right)})}_{\varphi }& X_{14,14}^{\left(2\right)}+X_{10,10}^{\left(3\right)}\\ -{(X_{8,15}^{\left(2\right)}+X_{6,11}^{\left(3\right)})}_{\varphi }& -{(X_{9,15}^{\left(2\right)}+X_{7,11}^{\left(3\right)})}_{\varphi }& -{(X_{10,15}^{\left(2\right)}+X_{8,11}^{\left(3\right)})}_{\varphi }& -{\left(X_{11,15}^{\left(2\right)}\right)}_{\varphi }& -{(X_{13,15}^{\left(2\right)}+X_{9,11}^{\left(3\right)})}_{\varphi }& -{(X_{14,15}^{\left(2\right)}+X_{10,11}^{\left(3\right)})}_{\varphi }\\ -{(X_{8,16}^{\left(2\right)}+X_{6,12}^{\left(3\right)})}_{\varphi }& -{(X_{9,16}^{\left(2\right)}+X_{7,12}^{\left(3\right)})}_{\varphi }& -{(X_{10,16}^{\left(2\right)}+X_{8,12}^{\left(3\right)})}_{\varphi }& -{\left(X_{11,16}^{\left(2\right)}\right)}_{\varphi }& -{(X_{13,16}^{\left(2\right)}+X_{9,12}^{\left(3\right)})}_{\varphi }& -{(X_{14,16}^{\left(2\right)}+X_{10,12}^{\left(3\right)})}_{\varphi }\\ -{\left(X_{8,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{9,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{10,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{11,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{13,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{14,17}^{\left(2\right)}\right)}_{\varphi }\\ -{\left(X_{6,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{8,13}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{9,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,13}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{6,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{8,14}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{9,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,14}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{6,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{8,15}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{9,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,15}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{6,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{8,16}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{9,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,16}^{\left(3\right)}\right)}_{\varphi }\\ 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill D_3^{\left(2\right)}=\left(\begin{array}{c}\hfill \begin{array}{cccccccc}{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_{1,14}^{\left(3\right)}& X_{1,15}^{\left(3\right)}& X_{1,16}^{\left(3\right)}& 0\\ 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_{2,14}^{\left(3\right)}& X_{2,15}^{\left(3\right)}& X_{2,16}^{\left(3\right)}& 0\\ 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_{3,14}^{\left(3\right)}& X_{3,15}^{\left(3\right)}& X_{3,16}^{\left(3\right)}& 0\\ 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_{4,14}^{\left(3\right)}& X_{4,15}^{\left(3\right)}& X_{4,16}^{\left(3\right)}& 0\\ X_{5,15}^{\left(2\right)}& X_{5,16}^{\left(2\right)}& X_{5,17}^{\left(2\right)}& 0& 0& 0& 0& 0\\ X_{7,15}^{\left(2\right)}+X_{5,11}^{\left(3\right)}& X_{7,16}^{\left(2\right)}+X_{5,12}^{\left(3\right)}& X_{7,17}^{\left(2\right)}& X_{5,13}^{\left(3\right)}& X_{5,14}^{\left(3\right)}& X_{5,15}^{\left(3\right)}& X_{5,16}^{\left(3\right)}& 0\\ 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_{6,14}^{\left(3\right)}& X_{6,15}^{\left(3\right)}& X_{6,16}^{\left(3\right)}& 0\\ 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_{7,14}^{\left(3\right)}& X_{7,15}^{\left(3\right)}& X_{7,16}^{\left(3\right)}& 0\\ 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_{8,14}^{\left(3\right)}& X_{8,15}^{\left(3\right)}& X_{8,16}^{\left(3\right)}& 0\\ X_{11,15}^{\left(2\right)}& X_{11,16}^{\left(2\right)}& X_{11,17}^{\left(2\right)}& 0& 0& 0& 0& 0\\ X_{13,15}^{\left(2\right)}+X_{9,11}^{\left(3\right)}& X_{13,16}^{\left(2\right)}+X_{9,12}^{\left(3\right)}& X_{13,17}^{\left(2\right)}& X_{9,13}^{\left(3\right)}& X_{9,14}^{\left(3\right)}& X_{9,15}^{\left(3\right)}& X_{9,16}^{\left(3\right)}& 0\\ X_{14,15}^{\left(2\right)}+X_{10,11}^{\left(3\right)}& X_{14,16}^{\left(2\right)}+X_{10,12}^{\left(3\right)}& X_{14,17}^{\left(2\right)}& X_{10,13}^{\left(3\right)}& X_{10,14}^{\left(3\right)}& X_{10,15}^{\left(3\right)}& X_{10,16}^{\left(3\right)}& 0\\ 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_{11,14}^{\left(3\right)}& X_{11,15}^{\left(3\right)}& X_{11,16}^{\left(3\right)}& 0\\ -{(X_{15,16}^{\left(2\right)}+X_{11,12}^{\left(3\right)})}_{\varphi }& 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_{12,14}^{\left(3\right)}& X_{12,15}^{\left(3\right)}& X_{12,16}^{\left(3\right)}& 0\\ -{\left(X_{15,17}^{\left(2\right)}\right)}_{\varphi }& -{\left(X_{16,17}^{\left(2\right)}\right)}_{\varphi }& X_{17,17}^{\left(2\right)}& 0& 0& 0& 0& 0\\ -{\left(X_{11,13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,13}^{\left(3\right)}\right)}_{\varphi }& 0& X_{13,13}^{\left(3\right)}& X_{13,14}^{\left(3\right)}& X_{13,15}^{\left(3\right)}& X_{13,16}^{\left(3\right)}& 0\\ -{\left(X_{11,14}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,14}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{13,14}^{\left(3\right)}\right)}_{\varphi }& X_{14,14}^{\left(3\right)}& X_{14,15}^{\left(3\right)}& X_{14,16}^{\left(3\right)}& 0\\ -{\left(X_{11,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,15}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{13,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{14,15}^{\left(3\right)}\right)}_{\varphi }& X_{15,15}^{\left(3\right)}& X_{15,16}^{\left(3\right)}& 0\\ -{\left(X_{11,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{12,16}^{\left(3\right)}\right)}_{\varphi }& 0& -{\left(X_{13,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{14,16}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{15,16}^{\left(3\right)}\right)}_{\varphi }& X_{16,16}^{\left(3\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}\right).\end{array}$$

Substituting $\widehat{X_3}$ and $\widehat{X_4}$ into the third equation in (4) yields

$$\begin{array}{c}\hfill {\left(D_{ij}^{\left(3\right)}\right)}_{18\times 18}=(D_1^{\left(3\right)},D_2^{\left(3\right)},D_3^{\left(3\right)}),\end{array}$$

(7)

where

$$\begin{array}{c}\hfill D_1^{\left(3\right)}=\left(\begin{array}{c}\hfill \begin{array}{cccccc}{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_{15}^{\left(3\right)}+X_{13}^{\left(4\right)}& X_{16}^{\left(3\right)}+X_{14}^{\left(4\right)}& X_{17}^{\left(3\right)}\\ -{(X_{12}^{\left(3\right)}+X_{12}^{\left(4\right)})}_{\varphi }& X_{22}^{\left(3\right)}+X_{22}^{\left(4\right)}& X_{23}^{\left(3\right)}& X_{25}^{\left(3\right)}+X_{23}^{\left(4\right)}& X_{26}^{\left(3\right)}+X_{24}^{\left(4\right)}& X_{27}^{\left(3\right)}\\ -{\left(X_{13}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{23}^{\left(3\right)}\right)}_{\varphi }& X_{33}^{\left(3\right)}& X_{35}^{\left(3\right)}& X_{36}^{\left(3\right)}& X_{37}^{\left(3\right)}\\ -{(X_{15}^{\left(3\right)}+X_{13}^{\left(4\right)})}_{\varphi }& -{(X_{25}^{\left(3\right)}+X_{23}^{\left(4\right)})}_{\varphi }& -{\left(X_{35}^{\left(3\right)}\right)}_{\varphi }& X_{55}^{\left(3\right)}+X_{33}^{\left(4\right)}& X_{56}^{\left(3\right)}+X_{34}^{\left(4\right)}& X_{57}^{\left(3\right)}\\ -{(X_{16}^{\left(3\right)}+X_{14}^{\left(4\right)})}_{\varphi }& -{(X_{26}^{\left(3\right)}+X_{24}^{\left(4\right)})}_{\varphi }& -{\left(X_{36}^{\left(3\right)}\right)}_{\varphi }& -{(X_{56}^{\left(3\right)}+X_{34}^{\left(4\right)})}_{\varphi }& X_{66}^{\left(3\right)}+X_{44}^{\left(4\right)}& X_{67}^{\left(3\right)}\\ -{\left(X_{17}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{27}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{37}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{57}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{67}^{\left(3\right)}\right)}_{\varphi }& X_{77}^{\left(3\right)}\\ -{(X_{19}^{\left(3\right)}+X_{15}^{\left(4\right)})}_{\varphi }& -{(X_{29}^{\left(3\right)}+X_{25}^{\left(4\right)})}_{\varphi }& -{\left(X_{39}^{\left(3\right)}\right)}_{\varphi }& -{(X_{59}^{\left(3\right)}+X_{35}^{\left(4\right)})}_{\varphi }& -{(X_{69}^{\left(3\right)}+X_{45}^{\left(4\right)})}_{\varphi }& -{\left(X_{79}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{1,10}^{\left(3\right)}+X_{16}^{\left(4\right)})}_{\varphi }& -{(X_{2,10}^{\left(3\right)}+X_{26}^{\left(4\right)})}_{\varphi }& -{\left(X_{3,10}^{\left(3\right)}\right)}_{\varphi }& -{(X_{5,10}^{\left(3\right)}+X_{36}^{\left(4\right)})}_{\varphi }& -{(X_{6,10}^{\left(3\right)}+X_{46}^{\left(4\right)})}_{\varphi }& -{\left(X_{7,10}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{5,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{6,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,11}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{1,13}^{\left(3\right)}+X_{17}^{\left(4\right)})}_{\varphi }& -{(X_{2,13}^{\left(3\right)}+X_{27}^{\left(4\right)})}_{\varphi }& -{\left(X_{3,13}^{\left(3\right)}\right)}_{\varphi }& -{(X_{5,13}^{\left(3\right)}+X_{37}^{\left(4\right)})}_{\varphi }& -{(X_{6,13}^{\left(3\right)}+X_{47}^{\left(4\right)})}_{\varphi }& -{\left(X_{7,13}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{1,14}^{\left(3\right)}+X_{18}^{\left(4\right)})}_{\varphi }& -{(X_{2,14}^{\left(3\right)}+X_{28}^{\left(4\right)})}_{\varphi }& -{\left(X_{3,14}^{\left(3\right)}\right)}_{\varphi }& -{(X_{5,14}^{\left(3\right)}+X_{38}^{\left(4\right)})}_{\varphi }& -{(X_{6,14}^{\left(3\right)}+X_{48}^{\left(4\right)})}_{\varphi }& -{\left(X_{7,14}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{5,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{6,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,15}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{1,17}^{\left(3\right)}+X_{19}^{\left(4\right)})}_{\varphi }& -{(X_{2,17}^{\left(3\right)}+X_{29}^{\left(4\right)})}_{\varphi }& -{\left(X_{3,17}^{\left(3\right)}\right)}_{\varphi }& -{(X_{5,17}^{\left(3\right)}+X_{39}^{\left(4\right)})}_{\varphi }& -{(X_{6,17}^{\left(3\right)}+X_{49}^{\left(4\right)})}_{\varphi }& -{\left(X_{7,17}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{1,18}^{\left(3\right)}+X_{1,10}^{\left(4\right)})}_{\varphi }& -{(X_{2,18}^{\left(3\right)}+X_{2,10}^{\left(4\right)})}_{\varphi }& -{\left(X_{3,18}^{\left(3\right)}\right)}_{\varphi }& -{(X_{5,18}^{\left(3\right)}+X_{3,10}^{\left(4\right)})}_{\varphi }& -{(X_{6,18}^{\left(3\right)}+X_{4,10}^{\left(4\right)})}_{\varphi }& -{\left(X_{7,18}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{2,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{3,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{5,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{6,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{7,19}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{1,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{2,11}^{\left(4\right)}\right)}_{\varphi }& 0& -{\left(X_{3,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{4,11}^{\left(4\right)}\right)}_{\varphi }& 0\\ -{\left(X_{1,12}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{2,12}^{\left(4\right)}\right)}_{\varphi }& 0& -{\left(X_{3,12}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{4,12}^{\left(4\right)}\right)}_{\varphi }& 0\\ 0& 0& 0& 0& 0& 0\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill D_2^{\left(3\right)}=\left(\begin{array}{c}\hfill \begin{array}{cccccc}{X}_{19}^{\left(3\right)}+X_{15}^{\left(4\right)}& X_{1,10}^{\left(3\right)}+X_{16}^{\left(4\right)}& X_{1,11}^{\left(3\right)}& X_{1,13}^{\left(3\right)}+X_{17}^{\left(4\right)}& X_{1,14}^{\left(3\right)}+X_{18}^{\left(4\right)}& X_{1,15}^{\left(3\right)}\\ X_{29}^{\left(3\right)}+X_{25}^{\left(4\right)}& X_{2,10}^{\left(3\right)}+X_{26}^{\left(4\right)}& X_{2,11}^{\left(3\right)}& X_{2,13}^{\left(3\right)}+X_{27}^{\left(4\right)}& X_{2,14}^{\left(3\right)}+X_{28}^{\left(4\right)}& X_{2,15}^{\left(3\right)}\\ X_{39}^{\left(3\right)}& X_{3,10}^{\left(3\right)}& X_{3,11}^{\left(3\right)}& X_{3,13}^{\left(3\right)}& X_{3,14}^{\left(3\right)}& X_{3,15}^{\left(3\right)}\\ X_{59}^{\left(3\right)}+X_{35}^{\left(4\right)}& X_{5,10}^{\left(3\right)}+X_{36}^{\left(4\right)}& X_{5,11}^{\left(3\right)}& X_{5,13}^{\left(3\right)}+X_{37}^{\left(4\right)}& X_{5,14}^{\left(3\right)}+X_{38}^{\left(4\right)}& X_{5,15}^{\left(3\right)}\\ X_{69}^{\left(3\right)}+X_{45}^{\left(4\right)}& X_{6,10}^{\left(3\right)}+X_{46}^{\left(4\right)}& X_{6,11}^{\left(3\right)}& X_{6,13}^{\left(3\right)}+X_{47}^{\left(4\right)}& X_{6,14}^{\left(3\right)}+X_{48}^{\left(4\right)}& X_{6,15}^{\left(3\right)}\\ X_{79}^{\left(3\right)}& X_{7,10}^{\left(3\right)}& X_{7,11}^{\left(3\right)}& X_{7,13}^{\left(3\right)}& X_{7,14}^{\left(3\right)}& X_{7,15}^{\left(3\right)}\\ X_{99}^{\left(3\right)}+X_{55}^{\left(4\right)}& X_{9,10}^{\left(3\right)}+X_{56}^{\left(4\right)}& X_{9,11}^{\left(3\right)}& X_{9,13}^{\left(3\right)}+X_{57}^{\left(4\right)}& X_{9,14}^{\left(3\right)}+X_{58}^{\left(4\right)}& X_{9,15}^{\left(3\right)}\\ -{(X_{9,10}^{\left(3\right)}+X_{56}^{\left(4\right)})}_{\varphi }& X_{10,10}^{\left(3\right)}+X_{66}^{\left(4\right)}& X_{10,11}^{\left(3\right)}& X_{10,13}^{\left(3\right)}+X_{67}^{\left(4\right)}& X_{10,14}^{\left(3\right)}+X_{68}^{\left(4\right)}& X_{10,15}^{\left(3\right)}\\ -{\left(X_{9,11}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,11}^{\left(3\right)}\right)}_{\varphi }& X_{11,11}^{\left(3\right)}& X_{11,13}^{\left(3\right)}& X_{11,14}^{\left(3\right)}& X_{11,15}^{\left(3\right)}\\ -{(X_{9,13}^{\left(3\right)}+X_{57}^{\left(4\right)})}_{\varphi }& -{(X_{10,13}^{\left(3\right)}+X_{67}^{\left(4\right)})}_{\varphi }& -{\left(X_{11,13}^{\left(3\right)}\right)}_{\varphi }& X_{13,13}^{\left(3\right)}+X_{77}^{\left(4\right)}& X_{13,14}^{\left(3\right)}+X_{78}^{\left(4\right)}& X_{13,15}^{\left(3\right)}\\ -{(X_{9,14}^{\left(3\right)}+X_{58}^{\left(4\right)})}_{\varphi }& -{(X_{10,14}^{\left(3\right)}+X_{68}^{\left(4\right)})}_{\varphi }& -{\left(X_{11,14}^{\left(3\right)}\right)}_{\varphi }& -{(X_{13,14}^{\left(3\right)}+X_{78}^{\left(4\right)})}_{\varphi }& X_{14,14}^{\left(3\right)}+X_{88}^{\left(4\right)}& X_{14,15}^{\left(3\right)}\\ -{\left(X_{9,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{11,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{13,15}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{14,15}^{\left(3\right)}\right)}_{\varphi }& X_{15,15}^{\left(3\right)}\\ -{(X_{9,17}^{\left(3\right)}+X_{59}^{\left(4\right)})}_{\varphi }& -{(X_{10,17}^{\left(3\right)}+X_{69}^{\left(4\right)})}_{\varphi }& -{\left(X_{11,17}^{\left(3\right)}\right)}_{\varphi }& -{(X_{13,17}^{\left(3\right)}+X_{79}^{\left(4\right)})}_{\varphi }& -{(X_{14,17}^{\left(3\right)}+X_{89}^{\left(4\right)})}_{\varphi }& -{\left(X_{15,17}^{\left(3\right)}\right)}_{\varphi }\\ -{(X_{9,18}^{\left(3\right)}+X_{5,10}^{\left(4\right)})}_{\varphi }& -{(X_{10,18}^{\left(3\right)}+X_{6,10}^{\left(4\right)})}_{\varphi }& -{\left(X_{11,18}^{\left(3\right)}\right)}_{\varphi }& -{(X_{13,18}^{\left(3\right)}+X_{7,10}^{\left(4\right)})}_{\varphi }& -{(X_{14,18}^{\left(3\right)}+X_{8,10}^{\left(4\right)})}_{\varphi }& -{\left(X_{15,18}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{9,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{10,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{11,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{13,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{14,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{15,19}^{\left(3\right)}\right)}_{\varphi }\\ -{\left(X_{5,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{6,11}^{\left(4\right)}\right)}_{\varphi }& 0& -{\left(X_{7,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{8,11}^{\left(4\right)}\right)}_{\varphi }& 0\\ -{\left(X_{5,12}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{6,12}^{\left(4\right)}\right)}_{\varphi }& 0& -{\left(X_{7,12}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{8,12}^{\left(4\right)}\right)}_{\varphi }& 0\\ 0& 0& 0& 0& 0& 0\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill D_3^{\left(3\right)}=\begin{array}{c}\hfill \left(\begin{array}{cccccc}{X}_{1,17}^{\left(3\right)}+X_{19}^{\left(4\right)}& X_{1,18}^{\left(3\right)}+X_{1,10}^{\left(4\right)}& X_{1,19}^{\left(3\right)}& X_{1,11}^{\left(4\right)}& X_{1,12}^{\left(4\right)}& 0\\ X_{2,17}^{\left(3\right)}+X_{29}^{\left(4\right)}& X_{2,18}^{\left(3\right)}+X_{2,10}^{\left(4\right)}& X_{2,19}^{\left(3\right)}& X_{2,11}^{\left(4\right)}& X_{2,12}^{\left(4\right)}& 0\\ X_{3,17}^{\left(3\right)}& X_{3,18}^{\left(3\right)}& X_{3,19}^{\left(3\right)}& 0& 0& 0\\ X_{5,17}^{\left(3\right)}+X_{39}^{\left(4\right)}& X_{5,18}^{\left(3\right)}+X_{3,10}^{\left(4\right)}& X_{5,19}^{\left(3\right)}& X_{3,11}^{\left(4\right)}& X_{3,12}^{\left(4\right)}& 0\\ X_{6,17}^{\left(3\right)}+X_{49}^{\left(4\right)}& X_{6,18}^{\left(3\right)}+X_{4,10}^{\left(4\right)}& X_{6,19}^{\left(3\right)}& X_{4,11}^{\left(4\right)}& X_{4,12}^{\left(4\right)}& 0\\ X_{7,17}^{\left(3\right)}& X_{7,18}^{\left(3\right)}& X_{7,19}^{\left(3\right)}& 0& 0& 0\\ X_{9,17}^{\left(3\right)}+X_{59}^{\left(4\right)}& X_{9,18}^{\left(3\right)}+X_{5,10}^{\left(4\right)}& X_{9,19}^{\left(3\right)}& X_{5,11}^{\left(4\right)}& X_{5,12}^{\left(4\right)}& 0\\ X_{10,17}^{\left(3\right)}+X_{69}^{\left(4\right)}& X_{10,18}^{\left(3\right)}+X_{6,10}^{\left(4\right)}& X_{10,19}^{\left(3\right)}& X_{6,11}^{\left(4\right)}& X_{6,12}^{\left(4\right)}& 0\\ X_{11,17}^{\left(3\right)}& X_{11,18}^{\left(3\right)}& X_{11,19}^{\left(3\right)}& 0& 0& 0\\ X_{13,17}^{\left(3\right)}+X_{79}^{\left(4\right)}& X_{13,18}^{\left(3\right)}+X_{7,10}^{\left(4\right)}& X_{13,19}^{\left(3\right)}& X_{7,11}^{\left(4\right)}& X_{7,12}^{\left(4\right)}& 0\\ X_{14,17}^{\left(3\right)}+X_{89}^{\left(4\right)}& X_{14,18}^{\left(3\right)}+X_{8,10}^{\left(4\right)}& X_{14,19}^{\left(3\right)}& X_{8,11}^{\left(4\right)}& X_{8,12}^{\left(4\right)}& 0\\ X_{15,17}^{\left(3\right)}& X_{15,18}^{\left(3\right)}& X_{15,19}^{\left(3\right)}& 0& 0& 0\\ X_{17,17}^{\left(3\right)}+X_{99}^{\left(4\right)}& X_{17,18}^{\left(3\right)}+X_{9,10}^{\left(4\right)}& X_{17,19}^{\left(3\right)}& X_{9,11}^{\left(4\right)}& X_{9,12}^{\left(4\right)}& 0\\ -{(X_{17,18}^{\left(3\right)}+X_{9,10}^{\left(4\right)})}_{\varphi }& X_{18,18}^{\left(3\right)}+X_{10,10}^{\left(4\right)}& X_{18,19}^{\left(3\right)}& X_{10,11}^{\left(4\right)}& X_{10,12}^{\left(4\right)}& 0\\ -{\left(X_{17,19}^{\left(3\right)}\right)}_{\varphi }& -{\left(X_{18,19}^{\left(3\right)}\right)}_{\varphi }& X_{19,19}^{\left(3\right)}& 0& 0& 0\\ -{\left(X_{9,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{10,11}^{\left(4\right)}\right)}_{\varphi }& 0& X_{11,11}^{\left(4\right)}& X_{11,12}^{\left(4\right)}& 0\\ -{\left(X_{9,12}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{10,12}^{\left(4\right)}\right)}_{\varphi }& 0& -{\left(X_{11,12}^{\left(4\right)}\right)}_{\varphi }& X_{12,12}^{\left(4\right)}& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right).\end{array}\end{array}$$

Substituting $\widehat{X_4}$ into the fourth equation in (4) yields

$$\begin{array}{c}\hfill {\left(D_{ij}^{\left(4\right)}\right)}_{8\times 8}=\left(\begin{array}{c}\hfill \begin{array}{cccccccc}{X}_{11}^{\left(4\right)}& X_{13}^{\left(4\right)}& X_{15}^{\left(4\right)}& X_{17}^{\left(4\right)}& X_{19}^{\left(4\right)}& X_{1,11}^{\left(4\right)}& X_{1,13}^{\left(4\right)}& 0\\ -{\left(X_{13}^{\left(4\right)}\right)}_{\varphi }& X_{33}^{\left(4\right)}& X_{35}^{\left(4\right)}& X_{37}^{\left(4\right)}& X_{39}^{\left(4\right)}& X_{3,11}^{\left(4\right)}& X_{3,13}^{\left(4\right)}& 0\\ -{\left(X_{15}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{35}^{\left(4\right)}\right)}_{\varphi }& X_{55}^{\left(4\right)}& X_{57}^{\left(4\right)}& X_{59}^{\left(4\right)}& X_{5,11}^{\left(4\right)}& X_{5,13}^{\left(4\right)}& 0\\ -{\left(X_{17}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{37}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{57}^{\left(4\right)}\right)}_{\varphi }& X_{77}^{\left(4\right)}& X_{79}^{\left(4\right)}& X_{7,11}^{\left(4\right)}& X_{7,13}^{\left(4\right)}& 0\\ -{\left(X_{19}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{39}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{59}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{79}^{\left(4\right)}\right)}_{\varphi }& X_{99}^{\left(4\right)}& X_{9,11}^{\left(4\right)}& X_{9,13}^{\left(4\right)}& 0\\ -{\left(X_{1,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{3,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{5,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{7,11}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{9,11}^{\left(4\right)}\right)}_{\varphi }& X_{11,11}^{\left(4\right)}& X_{11,13}^{\left(4\right)}& 0\\ -{\left(X_{1,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{3,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{5,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{7,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{9,13}^{\left(4\right)}\right)}_{\varphi }& -{\left(X_{11,13}^{\left(4\right)}\right)}_{\varphi }& X_{13,13}^{\left(4\right)}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}\right).\end{array}$$

(8)

Hence, the system of (1) has a $\varphi$-skew-Hermitian solution $(X_1,X_2,X_3,X_4)$ if and only if Equation (4) has a $\varphi$-skew-Hermitian solution. Note that (5)–(8) are consistent if and only if

$$\begin{array}{c}\hfill \left(\begin{array}{c}{D}_{14,1}^{\left(1\right)}\\ D_{14,2}^{\left(1\right)}\\ ⋮\\ D_{14,14}^{\left(1\right)}\end{array}\right)=0,\left(\begin{array}{c}{D}_{20,1}^{\left(2\right)}\\ D_{20,2}^{\left(2\right)}\\ ⋮\\ D_{20,20}^{\left(2\right)}\end{array}\right)=0,\left(\begin{array}{c}{D}_{18,1}^{\left(3\right)}\\ D_{18,2}^{\left(3\right)}\\ ⋮\\ D_{18,18}^{\left(3\right)}\end{array}\right)=0,\left(\begin{array}{c}{D}_{81}^{\left(4\right)}\\ D_{82}^{\left(4\right)}\\ ⋮\\ D_{88}^{\left(4\right)}\end{array}\right)=0,\end{array}$$

(9)

$$\begin{array}{c}\hfill \left(\begin{array}{c}{D}_{1,14}^{\left(1\right)}\\ D_{2,14}^{\left(1\right)}\\ ⋮\\ D_{14,14}^{\left(1\right)}\end{array}\right)=0,\left(\begin{array}{c}{D}_{1,20}^{\left(2\right)}\\ D_{2,20}^{\left(2\right)}\\ ⋮\\ D_{20,20}^{\left(2\right)}\end{array}\right)=0,\left(\begin{array}{c}{D}_{1,18}^{\left(3\right)}\\ D_{2,18}^{\left(3\right)}\\ ⋮\\ D_{18,18}^{\left(3\right)}\end{array}\right)=0,\left(\begin{array}{c}{D}_{18}^{\left(4\right)}\\ D_{28}^{\left(4\right)}\\ ⋮\\ D_{88}^{\left(4\right)}\end{array}\right)=0,\end{array}$$

(10)

$$\begin{array}{c}\hfill \left(\begin{array}{c}{D}_{87}^{\left(1\right)}\\ D_{97}^{\left(1\right)}\\ ⋮\\ D_{14,7}^{\left(1\right)}\end{array}\right)=0,\end{array}$$

(11)

$$\begin{array}{c}\hfill \left(\begin{array}{c}{D}_{16,5}^{\left(2\right)}\\ D_{17,5}^{\left(2\right)}\\ ⋮\\ D_{20,5}^{\left(2\right)}\end{array}\right)=0,\left(\begin{array}{c}{D}_{16,10}^{\left(2\right)}\\ D_{17,10}^{\left(2\right)}\\ ⋮\\ D_{20,10}^{\left(2\right)}\end{array}\right)=0,\left(\begin{array}{c}{D}_{16,15}^{\left(2\right)}\\ D_{17,15}^{\left(2\right)}\\ ⋮\\ D_{20,15}^{\left(2\right)}\end{array}\right)=0,\end{array}$$

(12)

$$\begin{array}{c}\hfill \left(\begin{array}{c}{D}_{16,3}^{\left(3\right)}\hfill \\ D_{17,3}^{\left(3\right)}\hfill \\ D_{18,3}^{\left(3\right)}\hfill \end{array}\right)=0,\left(\begin{array}{c}{D}_{16,6}^{\left(3\right)}\hfill \\ D_{17,6}^{\left(3\right)}\hfill \\ D_{18,6}^{\left(3\right)}\hfill \end{array}\right)=0,\left(\begin{array}{c}{D}_{16,9}^{\left(3\right)}\hfill \\ D_{17,9}^{\left(3\right)}\hfill \\ D_{18,9}^{\left(3\right)}\hfill \end{array}\right)=0,\left(\begin{array}{c}{D}_{16,12}^{\left(3\right)}\hfill \\ D_{17,12}^{\left(3\right)}\hfill \\ D_{18,12}^{\left(3\right)}\hfill \end{array}\right)=0,\left(\begin{array}{c}{D}_{16,15}^{\left(3\right)}\hfill \\ D_{17,15}^{\left(3\right)}\hfill \\ D_{18,15}^{\left(3\right)}\hfill \end{array}\right)=0,\end{array}$$

(13)

$$\begin{array}{c}\hfill D_{12,1}^{\left(1\right)}=D_{10,1}^{\left(2\right)},D_{12,2}^{\left(1\right)}=D_{10,2}^{\left(2\right)},D_{12,3}^{\left(1\right)}=D_{10,3}^{\left(2\right)},D_{12,4}^{\left(1\right)}=D_{10,4}^{\left(2\right)},D_{12,5}^{\left(1\right)}=D_{10,5}^{\left(2\right)},\end{array}$$

$$\begin{array}{c}\hfill D_{12,8}^{\left(1\right)}=D_{10,6}^{\left(2\right)},D_{12,9}^{\left(1\right)}=D_{10,7}^{\left(2\right)},D_{12,10}^{\left(1\right)}=D_{10,8}^{\left(2\right)},D_{12,11}^{\left(1\right)}=D_{10,9}^{\left(2\right)},D_{12,12}^{\left(1\right)}=D_{10,10}^{\left(2\right)},\end{array}$$

(14)

$$\begin{array}{c}\hfill D_{18,1}^{\left(2\right)}=D_{12,1}^{\left(3\right)},D_{18,2}^{\left(2\right)}=D_{12,2}^{\left(3\right)},D_{18,3}^{\left(2\right)}=D_{12,3}^{\left(3\right)},D_{18,6}^{\left(2\right)}=D_{12,4}^{\left(3\right)},D_{18,7}^{\left(2\right)}=D_{12,5}^{\left(3\right)},D_{18,8}^{\left(2\right)}=D_{12,6}^{\left(3\right)},\end{array}$$

$$\begin{array}{c}\hfill D_{18,11}^{\left(2\right)}=D_{12,7}^{\left(3\right)},D_{18,12}^{\left(2\right)}=D_{12,8}^{\left(3\right)},D_{18,13}^{\left(2\right)}=D_{12,9}^{\left(3\right)},\end{array}$$

$$\begin{array}{c}\hfill D_{18,16}^{\left(2\right)}=D_{12,10}^{\left(3\right)},D_{18,17}^{\left(2\right)}=D_{12,11}^{\left(3\right)},D_{18,18}^{\left(2\right)}=D_{12,12}^{\left(3\right)},\end{array}$$

(15)

$$\begin{array}{c}\hfill D_{85}^{\left(1\right)}=D_{65}^{\left(2\right)},D_{95}^{\left(1\right)}=D_{75}^{\left(2\right)},D_{10,5}^{\left(1\right)}=D_{85}^{\left(2\right)},D_{11,5}^{\left(1\right)}=D_{95}^{\left(2\right)},D_{12,5}^{\left(1\right)}=D_{10,5}^{\left(2\right)},\end{array}$$

$$\begin{array}{c}\hfill D_{8,12}^{\left(1\right)}=D_{6,10}^{\left(2\right)},D_{9,12}^{\left(1\right)}=D_{7,10}^{\left(2\right)},D_{10,12}^{\left(1\right)}=D_{8,10}^{\left(2\right)},D_{11,12}^{\left(1\right)}=D_{9,10}^{\left(2\right)},D_{12,12}^{\left(1\right)}=D_{10,10}^{\left(2\right)},\end{array}$$

(16)

$$\begin{array}{c}\hfill D_{16,3}^{\left(2\right)}=D_{10,3}^{\left(3\right)},D_{17,3}^{\left(2\right)}=D_{11,3}^{\left(3\right)},D_{18,3}^{\left(2\right)}=D_{12,3}^{\left(3\right)},D_{16,8}^{\left(2\right)}=D_{10,6}^{\left(3\right)},D_{17,8}^{\left(2\right)}=D_{11,6}^{\left(3\right)},D_{18,8}^{\left(2\right)}=D_{12,6}^{\left(3\right)},\end{array}$$

$$\begin{array}{c}\hfill D_{16,13}^{\left(2\right)}=D_{10,9}^{\left(3\right)},D_{17,13}^{\left(2\right)}=D_{11,9}^{\left(3\right)},D_{18,13}^{\left(2\right)}=D_{12,9}^{\left(3\right)},\end{array}$$

$$\begin{array}{c}\hfill D_{16,18}^{\left(2\right)}=D_{10,12}^{\left(3\right)},D_{17,18}^{\left(2\right)}=D_{11,12}^{\left(3\right)},D_{18,18}^{\left(2\right)}=D_{12,12}^{\left(3\right)},\end{array}$$

(17)

$$\begin{array}{c}\hfill D_{16,1}^{\left(3\right)}=D_{61}^{\left(4\right)},D_{16,4}^{\left(3\right)}=D_{62}^{\left(4\right)},D_{16,7}^{\left(3\right)}=D_{63}^{\left(4\right)},D_{16,10}^{\left(3\right)}=D_{64}^{\left(4\right)},D_{16,13}^{\left(3\right)}=D_{65}^{\left(4\right)},D_{16,16}^{\left(3\right)}=D_{66}^{\left(4\right)},\end{array}$$

(18)

$$\begin{array}{c}\hfill D_{10,1}^{\left(1\right)}+D_{61}^{\left(3\right)}=D_{81}^{\left(2\right)},D_{10,2}^{\left(1\right)}+D_{62}^{\left(3\right)}=D_{82}^{\left(2\right)},D_{10,3}^{\left(1\right)}+D_{63}^{\left(3\right)}=D_{83}^{\left(2\right)},\end{array}$$

$$\begin{array}{c}\hfill D_{10,8}^{\left(1\right)}+D_{64}^{\left(3\right)}=D_{86}^{\left(2\right)},D_{10,9}^{\left(1\right)}+D_{65}^{\left(3\right)}=D_{87}^{\left(2\right)},D_{10,10}^{\left(1\right)}+D_{66}^{\left(3\right)}=D_{88}^{\left(2\right)},\end{array}$$

(19)

$$\begin{array}{c}\hfill D_{83}^{\left(1\right)}+D_{43}^{\left(3\right)}=D_{63}^{\left(2\right)},D_{93}^{\left(1\right)}+D_{53}^{\left(3\right)}=D_{73}^{\left(2\right)},D_{8,10}^{\left(1\right)}+D_{46}^{\left(3\right)}=D_{68}^{\left(2\right)},D_{9,10}^{\left(1\right)}+D_{56}^{\left(3\right)}=D_{78}^{\left(2\right)},\end{array}$$

(20)

$$\begin{array}{c}\hfill D_{16,1}^{\left(2\right)}+D_{41}^{\left(4\right)}=D_{10,1}^{\left(3\right)},D_{16,6}^{\left(2\right)}+D_{42}^{\left(4\right)}=D_{10,4}^{\left(3\right)},\end{array}$$

$$\begin{array}{c}\hfill D_{16,11}^{\left(2\right)}+D_{43}^{\left(4\right)}=D_{10,7}^{\left(3\right)},D_{16,16}^{\left(2\right)}+D_{44}^{\left(4\right)}=D_{10,10}^{\left(3\right)},\end{array}$$

(21)

$$\begin{array}{c}\hfill D_{81}^{\left(1\right)}+D_{41}^{\left(3\right)}=D_{61}^{\left(2\right)}+D_{21}^{\left(4\right)},D_{88}^{\left(1\right)}+D_{44}^{\left(3\right)}=D_{66}^{\left(2\right)}+D_{22}^{\left(4\right)}.\end{array}$$

(22)

Based on the above analysis, we have the following conclusions:

Theorem 1. 

The system (1) has a ϕ-skew-Hermitian solution ($X_1$, $X_2$, $X_3$, $X_4$) if and only if the Equations (9)–(22) hold.

The following theorem presents the solvability conditions to the system (1) in terms of rank.

Theorem 2. 

The system (1) has a ϕ-skew-Hermitian solution $(X_1,X_2,X_3,X_4)$ if and only if the ranks satisfy:

$$\begin{array}{c}\hfill r(A_i,C_i,B_i)=r(A_i,B_i),(i=1,2,3).\end{array}$$

(23)

$$\begin{array}{c}\hfill r\left(A_4,C_4\right)=r\left(A_4\right).\end{array}$$

(24)

$$\begin{array}{c}\hfill r\left(\begin{array}{cc}{A}_i& C_i\\ 0& {\left(B_i\right)}_{\varphi }\end{array}\right)=r\left(A_i\right)+r\left(B_i\right),(i=1,2,3).\end{array}$$

(25)

$$\begin{array}{c}\hfill r\left(\begin{array}{ccccc}{A}_j& C_j& B_j& 0& 0\\ 0& {\left(B_j\right)}_{\varphi }& 0& {\left(A_{j+1}\right)}_{\varphi }& 0\\ 0& 0& A_{j+1}& -C_{j+1}& B_{j+1}\end{array}\right)=r\left(\begin{array}{ccc}{A}_j& B_j& 0\\ 0& A_{j+1}& B_{j+1}\end{array}\right)+r\left(\begin{array}{c}{B}_j\\ A_{j+1}\end{array}\right)(j=1,2).\end{array}$$

(26)

$$\begin{array}{c}\hfill r\left(\begin{array}{cccc}{A}_j& C_j& B_j& 0\\ 0& {\left(B_j\right)}_{\varphi }& 0& {\left(A_{j+1}\right)}_{\varphi }\\ 0& 0& A_{j+1}& -C_{j+1}\\ 0& 0& 0& {\left(B_{j+1}\right)}_{\varphi }\end{array}\right)=r\left(\begin{array}{cc}{A}_j& B_j\\ 0& A_{j+1}\end{array}\right)+r\left(\begin{array}{cc}{B}_j& 0\\ A_{j+1}& B_{j+1}\end{array}\right)(j=1,2).\end{array}$$

(27)

$$\begin{array}{c}\hfill r\left(\begin{array}{cccc}{A}_3& C_3& B_3& 0\\ 0& {\left(B_3\right)}_{\varphi }& 0& {\left(A_4\right)}_{\varphi }\\ 0& 0& A_4& -C_4\end{array}\right)=r\left(\begin{array}{cc}{A}_3& B_3\\ 0& A_4\end{array}\right)+r\left(\begin{array}{c}{B}_3\\ A_4\end{array}\right).\end{array}$$

(28)

$$\begin{array}{c}\hfill r\begin{array}{c}\hfill \left(\begin{array}{ccccccc}{A}_1& C_1& B_1& 0& 0& 0& 0\\ 0& {\left(B_1\right)}_{\varphi }& 0& {\left(A_2\right)}_{\varphi }& 0& 0& 0\\ 0& 0& A_2& -C_2& B_2& 0& 0\\ 0& 0& 0& {\left(B_2\right)}_{\varphi }& 0& {\left(A_3\right)}_{\varphi }& 0\\ 0& 0& 0& 0& A_3& C_3& B_3\end{array}\right)\end{array}=r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\end{array}\right)+r\left(\begin{array}{cc}{B}_1& 0\\ A_2& B_2\\ 0& A_3\end{array}\right).\end{array}$$

(29)

$$\begin{array}{c}\hfill r\left(\begin{array}{cccccc}{A}_1& C_1& B_1& 0& 0& 0\\ 0& {\left(B_1\right)}_{\varphi }& 0& {\left(A_2\right)}_{\varphi }& 0& 0\\ 0& 0& A_2& -C_2& B_2& 0\\ 0& 0& 0& {\left(B_2\right)}_{\varphi }& 0& {\left(A_3\right)}_{\varphi }\\ 0& 0& 0& 0& A_3& C_3\\ 0& 0& 0& 0& 0& {\left(B_3\right)}_{\varphi }\end{array}\right)=r\left(\begin{array}{ccc}{A}_1& B_1& 0\\ 0& A_2& B_2\\ 0& 0& A_3\end{array}\right)+r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\end{array}\right).\end{array}$$

(30)

$$\begin{array}{c}\hfill r\left(\begin{array}{cccccc}{A}_2& C_2& B_2& 0& 0& 0\\ 0& {\left(B_2\right)}_{\varphi }& 0& {\left(A_3\right)}_{\varphi }& 0& 0\\ 0& 0& A_3& -C_3& B_3& 0\\ 0& 0& 0& {\left(B_3\right)}_{\varphi }& 0& {\left(A_4\right)}_{\varphi }\\ 0& 0& 0& 0& A_4& C_4\end{array}\right)=r\left(\begin{array}{ccc}{A}_2& B_2& 0\\ 0& A_3& B_3\\ 0& 0& A_4\end{array}\right)+r\left(\begin{array}{cc}{B}_2& 0\\ A_3& B_3\\ 0& A_4\end{array}\right).\end{array}$$

(31)

$$\begin{array}{c}\hfill r\left(\begin{array}{cccccccc}{A}_1& C_1& B_1& 0& 0& 0& 0& 0\\ 0& {\left(B_1\right)}_{\varphi }& 0& {\left(A_2\right)}_{\varphi }& 0& 0& 0& 0\\ 0& 0& A_2& -C_2& B_2& 0& 0& 0\\ 0& 0& 0& {\left(B_2\right)}_{\varphi }& 0& {\left(A_3\right)}_{\varphi }& 0& 0\\ 0& 0& 0& 0& A_3& C_3& B_3& 0\\ 0& 0& 0& 0& 0& {\left(B_3\right)}_{\varphi }& 0& {\left(A_4\right)}_{\varphi }\\ 0& 0& 0& 0& 0& 0& A_4& -C_4\end{array}\right)=r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\\ 0& 0& 0& A_4\end{array}\right)+r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\\ 0& 0& A_4\end{array}\right).\end{array}$$

(32)

Proof. 

According to the structure of the matrix, Lemma 1, Theorem 1, and the elementary transformation of the row and column of the matrix, we have

$$r(A_i,C_i,B_i)=r(A_i,B_i)$$

$$\iff r(\widehat{P_i}{A}_i\widehat{T_i},\widehat{P_i}{C}_i{\left(\widehat{P_i}\right)}_{\varphi },\widehat{P_i}{B}_i\widehat{T_{i+1}})=r(\widehat{P_i}{A}_i\widehat{T_i},\widehat{P_i}{B}_i\widehat{T_{i+1}})$$

$$\iff r(S_{a_i},D^{\left(i\right)},S_{b_i})=r(S_{a_i},S_{b_i})$$

$$\iff D_{14,j}^{\left(1\right)}=0,(j=1,\cdots ,14),D_{20,j}^{\left(2\right)}=0,(j=1,\cdots ,20),D_{18,j}^{\left(3\right)}=0,(j=1,\cdots ,18).$$

$$r(A_4,C_4)=r\left(A_4\right)$$

$$\iff r(\widehat{P_4}{A}_4\widehat{T_4},\widehat{P_4}{C}_4{\left(\widehat{P_4}\right)}_{\varphi })=r\left(\widehat{P_4}{A}_4\widehat{T_4}\right)$$

$$\iff r(S_{a_4},D^{\left(4\right)})=r\left(S_{a_4}\right)$$

$$\iff D_{8j}^{\left(4\right)}=0,(j=1,\cdots ,8).$$

$$r\left(\begin{array}{cc}{A}_1& C_1\\ 0& {\left(B_1\right)}_{\varphi }\end{array}\right)=r\left(A_1\right)+r\left(B_1\right)$$

$$\iff r\left(\begin{array}{cc}\widehat{P_1}{A}_1\widehat{T_1}& \widehat{P_1}{C}_1{\left(\widehat{P_1}\right)}_{\varphi }\\ 0& {\left(\widehat{T_2}\right)}_{\varphi }{\left(B_1\right)}_{\varphi }{\left(\widehat{P_1}\right)}_{\varphi }\end{array}\right)=r\left(\widehat{P_1}{A}_1\widehat{T_1}\right)+r\left(\widehat{P_1}{B}_1\widehat{T_2}\right)$$

$$\iff r\left(\begin{array}{cc}{S}_{a_1}& D^{\left(1\right)}\\ 0& {\left(S_{b_1}\right)}_{\varphi }\end{array}\right)=r\left(S_{a_1}\right)+r\left(S_{b_1}\right)$$

$$\iff D_{i,7}^{\left(1\right)}=0,(i=8,\cdots ,14),D_{i,14}^{\left(1\right)}=0,(i=8,\cdots ,14).$$

$$r\left(\begin{array}{cc}{A}_2& C_2\\ 0& {\left(B_2\right)}_{\varphi }\end{array}\right)=r\left(A_2\right)+r\left(B_2\right)$$

$$\iff r\left(\begin{array}{cc}\widehat{P_2}{A}_2\widehat{T_2}& \widehat{P_2}{C}_2{\left(\widehat{P_2}\right)}_{\varphi }\\ 0& {\left(\widehat{T_3}\right)}_{\varphi }{\left(B_2\right)}_{\varphi }{\left(\widehat{P_2}\right)}_{\varphi }\end{array}\right)=r\left(\widehat{P_2}{A}_2\widehat{T_2}\right)+r\left(\widehat{P_2}{B}_2\widehat{T_3}\right)$$

$$\iff r\left(\begin{array}{cc}{S}_{a_2}& D^{\left(2\right)}\\ 0& {\left(S_{b_2}\right)}_{\varphi }\end{array}\right)=r\left(S_{a_2}\right)+r\left(S_{b_2}\right)$$

$$\iff D_{i,5}^{\left(2\right)}=0,D_{i,10}^{\left(2\right)}=0,D_{i,15}^{\left(2\right)}=0,D_{i,20}^{\left(2\right)}=0,(i=16,\cdots ,20).$$

$$r\left(\begin{array}{cc}{A}_3& C_3\\ 0& {\left(B_3\right)}_{\varphi }\end{array}\right)=r\left(A_3\right)+r\left(B_3\right)$$

$$\iff r\left(\begin{array}{cc}\widehat{P_3}{A}_3\widehat{T_3}& \widehat{P_3}{C}_3{\left(\widehat{P_3}\right)}_{\varphi }\\ 0& {\left(\widehat{T_4}\right)}_{\varphi }{\left(B_3\right)}_{\varphi }{\left(\widehat{P_3}\right)}_{\varphi }\end{array}\right)=r\left(\widehat{P_3}{A}_3\widehat{T_3}\right)+r\left(\widehat{P_3}{B}_3\widehat{T_4}\right)$$

$$\iff r\left(\begin{array}{cc}{S}_{a_3}& D^{\left(3\right)}\\ 0& {\left(S_{b_3}\right)}_{\varphi }\end{array}\right)=r\left(S_{a_3}\right)+r\left(S_{b_3}\right)$$

$$\iff D_{i,3}^{\left(3\right)}=0,D_{i,6}^{\left(3\right)}=0,D_{i,9}^{\left(3\right)}=0,$$

$$D_{i,12}^{\left(3\right)}=0,D_{i,15}^{\left(3\right)}=0,D_{i,18}^{\left(3\right)}=0,(i=16,17,18).$$

$$r\left(\begin{array}{ccccc}{A}_1& C_1& B_1& 0& 0\\ 0& {\left(B_1\right)}_{\varphi }& 0& {\left(A_2\right)}_{\varphi }& 0\\ 0& 0& A_2& -C_2& B_2\end{array}\right)=r\left(\begin{array}{ccc}{A}_1& B_1& 0\\ 0& A_2& B_2\end{array}\right)+r\left(\begin{array}{c}{B}_1\\ A_2\end{array}\right)$$

$$\iff r\left(\begin{array}{ccccc}{S}_{a_1}& D^{\left(1\right)}& S_{b_1}& 0& 0\\ 0& {\left(S_{b_1}\right)}_{\varphi }& 0& {\left(S_{a_2}\right)}_{\varphi }& 0\\ 0& 0& S_{a_2}& -D^{\left(2\right)}& S_{b_2}\end{array}\right)=r\left(\begin{array}{ccc}{S}_{a_1}& S_{b_1}& 0\\ 0& S_{a_2}& S_{b_2}\end{array}\right)+r\left(\begin{array}{c}{S}_{b_1}\\ S_{a_2}\end{array}\right)$$

$$\iff D_{14,j}^{\left(1\right)}=0,(j=1,\cdots ,14),D_{12,7}^{\left(1\right)}=0,D_{12,14}^{\left(1\right)}=0,$$

$$D_{20,j}^{\left(2\right)}=0,(j=1,\cdots ,10,16,\cdots ,20),D_{10,j}^{\left(2\right)}=0,(j=16,\cdots ,20),$$

$$D_{12,1}^{\left(1\right)}=D_{10,1}^{\left(2\right)},D_{12,2}^{\left(1\right)}=D_{10,2}^{\left(2\right)},D_{12,3}^{\left(1\right)}=D_{10,3}^{\left(2\right)},D_{12,4}^{\left(1\right)}=D_{10,4}^{\left(2\right)},D_{12,5}^{\left(1\right)}=D_{10,5}^{\left(2\right)},$$

$$D_{12,8}^{\left(1\right)}=D_{10,6}^{\left(2\right)},D_{12,9}^{\left(1\right)}=D_{10,7}^{\left(2\right)},D_{12,10}^{\left(1\right)}=D_{10,8}^{\left(2\right)},D_{12,11}^{\left(1\right)}=D_{10,9}^{\left(2\right)},D_{12,12}^{\left(1\right)}=D_{10,10}^{\left(2\right)}.$$

$$r\left(\begin{array}{ccccc}{A}_2& C_2& B_2& 0& 0\\ 0& {\left(B_2\right)}_{\varphi }& 0& {\left(A_3\right)}_{\varphi }& 0\\ 0& 0& A_3& -C_3& B_3\end{array}\right)=r\left(\begin{array}{ccc}{A}_2& B_2& 0\\ 0& A_3& B_3\end{array}\right)+r\left(\begin{array}{c}{B}_2\\ A_3\end{array}\right)$$

$$\iff r\left(\begin{array}{ccccc}{S}_{a_2}& D^{\left(2\right)}& S_{b_2}& 0& 0\\ 0& {\left(S_{b_2}\right)}_{\varphi }& 0& {\left(S_{a_3}\right)}_{\varphi }& 0\\ 0& 0& S_{a_3}& -D^{\left(3\right)}& S_{b_3}\end{array}\right)=r\left(\begin{array}{ccc}{S}_{a_2}& S_{b_2}& 0\\ 0& S_{a_3}& S_{b_3}\end{array}\right)+r\left(\begin{array}{c}{S}_{b_2}\\ S_{a_3}\end{array}\right)$$

$$\iff D_{20,j}^{\left(2\right)}=0,(j=1,2,3,5,6,7,8,10,11,12,13,15,16,17,18,20),$$

$$D_{18,j}^{\left(3\right)}=0,(j=1,\cdots ,12,16,17,18),$$

$$D_{18,5}^{\left(2\right)}=0,D_{18,10}^{\left(2\right)}=0,D_{18,15}^{\left(2\right)}=0,D_{18,18}^{\left(2\right)}=0,$$

$$D_{12,16}^{\left(3\right)}=0,D_{12,17}^{\left(3\right)}=0,D_{12,18}^{\left(3\right)}=0,$$

$$D_{18,1}^{\left(2\right)}=D_{12,1}^{\left(3\right)},D_{18,2}^{\left(2\right)}=D_{12,2}^{\left(3\right)},D_{18,3}^{\left(2\right)}=D_{12,3}^{\left(3\right)},$$

$$D_{18,6}^{\left(2\right)}=D_{12,4}^{\left(3\right)},D_{18,7}^{\left(2\right)}=D_{12,5}^{\left(3\right)},D_{18,8}^{\left(2\right)}=D_{12,6}^{\left(3\right)},$$

$$D_{18,11}^{\left(2\right)}=D_{12,7}^{\left(3\right)},D_{18,12}^{\left(2\right)}=D_{12,8}^{\left(3\right)},D_{18,13}^{\left(2\right)}=D_{12,9}^{\left(3\right)},$$

$$D_{18,16}^{\left(2\right)}=D_{12,10}^{\left(3\right)},D_{18,17}^{\left(2\right)}=D_{12,11}^{\left(3\right)},D_{18,18}^{\left(2\right)}=D_{12,12}^{\left(3\right)}.$$

$$r\left(\begin{array}{cccc}{A}_1& C_1& B_1& 0\\ 0& {\left(B_1\right)}_{\varphi }& 0& {\left(A_2\right)}_{\varphi }\\ 0& 0& A_2& -C_2\\ 0& 0& 0& {\left(B_2\right)}_{\varphi }\end{array}\right)=r\left(\begin{array}{cc}{A}_1& B_1\\ 0& A_2\end{array}\right)+r\left(\begin{array}{cc}{B}_1& 0\\ A_2& B_2\end{array}\right)$$

$$\iff r\left(\begin{array}{cccc}{S}_{a_1}& D^{\left(1\right)}& S_{b_1}& 0\\ 0& {\left(S_{b_1}\right)}_{\varphi }& 0& {\left(S_{a_2}\right)}_{\varphi }\\ 0& 0& S_{a_2}& -D^{\left(2\right)}\\ 0& 0& 0& {\left(S_{b_2}\right)}_{\varphi }\end{array}\right)=r\left(\begin{array}{cc}{S}_{a_1}& S_{b_1}\\ 0& S_{a_2}\end{array}\right)+r\left(\begin{array}{cc}{S}_{b_1}& 0\\ S_{a_2}& S_{b_2}\end{array}\right)$$

$$\iff D_{i,14}^{\left(1\right)}=0,(i=8,\cdots ,14),D_{i,20}^{\left(2\right)}=0,(i=6,\cdots ,10,16,\cdots ,20),$$

$$D_{i,7}^{\left(1\right)}=0,(i=8,\cdots ,14),D_{14,5}^{\left(1\right)}=0,D_{14,12}^{\left(1\right)}=0,$$

$$D_{i,5}^{\left(2\right)}=0,D_{i,10}^{\left(2\right)}=0,(i=16,\cdots ,20),$$

$$D_{85}^{\left(1\right)}=D_{65}^{\left(2\right)},D_{95}^{\left(1\right)}=D_{75}^{\left(2\right)},D_{10,5}^{\left(1\right)}=D_{85}^{\left(2\right)},D_{11,5}^{\left(1\right)}=D_{95}^{\left(2\right)},D_{12,5}^{\left(1\right)}=D_{10,5}^{\left(2\right)},$$

$$D_{8,12}^{\left(1\right)}=D_{6,10}^{\left(2\right)},D_{9,12}^{\left(1\right)}=D_{7,10}^{\left(2\right)},D_{10,12}^{\left(1\right)}=D_{8,10}^{\left(2\right)},D_{11,12}^{\left(1\right)}=D_{9,10}^{\left(2\right)},D_{12,12}^{\left(1\right)}=D_{10,10}^{\left(2\right)}.$$

$$r\left(\begin{array}{cccc}{A}_2& C_2& B_2& 0\\ 0& {\left(B_2\right)}_{\varphi }& 0& {\left(A_3\right)}_{\varphi }\\ 0& 0& A_3& -C_3\\ 0& 0& 0& {\left(B_3\right)}_{\varphi }\end{array}\right)=r\left(\begin{array}{cc}{A}_2& B_2\\ 0& A_3\end{array}\right)+r\left(\begin{array}{cc}{B}_2& 0\\ A_3& B_3\end{array}\right)$$

$$\iff r\left(\begin{array}{cccc}{S}_{a_2}& D^{\left(2\right)}& S_{b_2}& 0\\ 0& {\left(S_{b_2}\right)}_{\varphi }& 0& {\left(S_{a_3}\right)}_{\varphi }\\ 0& 0& S_{a_3}& -D^{\left(3\right)}\\ 0& 0& 0& {\left(S_{b_3}\right)}_{\varphi }\end{array}\right)=r\left(\begin{array}{cc}{S}_{a_2}& S_{b_2}\\ 0& S_{a_3}\end{array}\right)+r\left(\begin{array}{cc}{S}_{b_2}& 0\\ S_{a_3}& S_{b_3}\end{array}\right)$$

$$\iff D_{i,5}^{\left(2\right)}=0,D_{i,10}^{\left(2\right)}=0,D_{i,15}^{\left(2\right)}=0,D_{i,20}^{\left(2\right)}=0,(i=16,17,18,20),$$

$$D_{i,3}^{\left(3\right)}=0,D_{i,6}^{\left(3\right)}=0,D_{i,9}^{\left(3\right)}=0,$$

$$D_{i,12}^{\left(3\right)}=0,(i=16,17,18),D_{i,18}^{\left(3\right)}=0,(i=10,11,12,16,17,18),$$

$$D_{16,3}^{\left(2\right)}=D_{10,3}^{\left(3\right)},D_{17,3}^{\left(2\right)}=D_{11,3}^{\left(3\right)},D_{18,3}^{\left(2\right)}=D_{12,3}^{\left(3\right)},$$

$$D_{16,8}^{\left(2\right)}=D_{10,6}^{\left(3\right)},D_{17,8}^{\left(2\right)}=D_{11,6}^{\left(3\right)},D_{18,8}^{\left(2\right)}=D_{12,6}^{\left(3\right)},$$

$$D_{16,13}^{\left(2\right)}=D_{10,9}^{\left(3\right)},D_{17,13}^{\left(2\right)}=D_{11,9}^{\left(3\right)},D_{18,13}^{\left(2\right)}=D_{12,9}^{\left(3\right)},$$

$$D_{16,18}^{\left(2\right)}=D_{10,12}^{\left(3\right)},D_{17,18}^{\left(2\right)}=D_{11,12}^{\left(3\right)},D_{18,18}^{\left(2\right)}=D_{12,12}^{\left(3\right)}.$$

$$r\left(\begin{array}{cccc}{A}_3& C_3& B_3& 0\\ 0& {\left(B_3\right)}_{\varphi }& 0& {\left(A_4\right)}_{\varphi }\\ 0& 0& A_4& -C_4\end{array}\right)=r\left(\begin{array}{cc}{A}_3& B_3\\ 0& A_4\end{array}\right)+r\left(\begin{array}{c}{B}_3\\ A_4\end{array}\right)$$

$$\iff r\left(\begin{array}{cccc}{S}_{a_3}& D^{\left(3\right)}& S_{b_3}& 0\\ 0& {\left(S_{b_3}\right)}_{\varphi }& 0& {\left(S_{a_4}\right)}_{\varphi }\\ 0& 0& S_{a_4}& -D^{\left(4\right)}\end{array}\right)=r\left(\begin{array}{cc}{S}_{a_3}& S_{b_3}\\ 0& S_{a_4}\end{array}\right)+r\left(\begin{array}{c}{S}_{b_3}\\ S_{a_4}\end{array}\right)$$

$$\iff D_{16,j}^{\left(3\right)}=0,(j=3,6,9,12,15,18),$$

$$D_{18,j}^{\left(3\right)}=0,(j=1,3,4,6,7,9,10,12,13,15,16,18),$$

$$D_{8j}^{\left(4\right)}=0,(j=1,\cdots ,6,8),D_{68}^{\left(4\right)}=0,$$

$$D_{16,1}^{\left(3\right)}=D_{61}^{\left(4\right)},D_{16,4}^{\left(3\right)}=D_{62}^{\left(4\right)},D_{16,7}^{\left(3\right)}=D_{63}^{\left(4\right)},$$

$$D_{16,10}^{\left(3\right)}=D_{64}^{\left(4\right)},D_{16,13}^{\left(3\right)}=D_{65}^{\left(4\right)},D_{16,16}^{\left(3\right)}=D_{66}^{\left(4\right)}.$$

$$r\left(\begin{array}{ccccccc}{A}_1& C_1& B_1& 0& 0& 0& 0\\ 0& {\left(B_1\right)}_{\varphi }& 0& {\left(A_2\right)}_{\varphi }& 0& 0& 0\\ 0& 0& A_2& -C_2& B_2& 0& 0\\ 0& 0& 0& {\left(B_2\right)}_{\varphi }& 0& {\left(A_3\right)}_{\varphi }& 0\\ 0& 0& 0& 0& A_3& C_3& B_3\end{array}\right)=r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\end{array}\right)+r\left(\begin{array}{cc}{B}_1& 0\\ A_2& B_2\\ 0& A_3\end{array}\right)$$

$$\iff r\left(\begin{array}{ccccccc}{S}_{a_1}& D^{\left(1\right)}& S_{b_1}& 0& 0& 0& 0\\ 0& {\left(S_{b_1}\right)}_{\varphi }& 0& {\left(S_{a_2}\right)}_{\varphi }& 0& 0& 0\\ 0& 0& S_{a_2}& -D^{\left(2\right)}& S_{a_2}& 0& 0\\ 0& 0& 0& {\left(S_{b_2}\right)}_{\varphi }& 0& {\left(S_{a_3}\right)}_{\varphi }& 0\\ 0& 0& 0& 0& S_{a_3}& D^{\left(3\right)}& S_{b_3}\end{array}\right)$$

$$=r\left(\begin{array}{cccc}{S}_{a_1}& S_{b_1}& 0& 0\\ 0& S_{a_2}& S_{b_2}& 0\\ 0& 0& S_{a_3}& S_{b_3}\end{array}\right)+r\left(\begin{array}{cc}{S}_{b_1}& 0\\ S_{a_2}& S_{b_2}\\ 0& S_{a_3}\end{array}\right)$$

$$\iff D_{14,j}^{\left(1\right)}=0,(j=1,2,3,5,7,8,9,10,12,14),D_{10,7}^{\left(1\right)}=0,$$

$$D_{12,7}^{\left(1\right)}=0,D_{10,14}^{\left(1\right)}=0,D_{12,14}^{\left(1\right)}=0,$$

$$D_{20,j}^{\left(2\right)}=0,(j=1,2,3,5,6,7,8,10,16,17,18,20),D_{8,20}^{\left(2\right)}=0,D_{10,20}^{\left(2\right)}=0,D_{18,20}^{\left(2\right)}=0,$$

$$D_{10,j}^{\left(2\right)}=0,(j=16,17,18),D_{18,5}^{\left(2\right)}=0,D_{18,10}^{\left(2\right)}=0,$$

$$D_{18,j}^{\left(3\right)}=0,(j=1,\cdots ,6,10,11,12,16,17,18),$$

$$D_{6,j}^{\left(3\right)}=0,(j=16,17,18),D_{12,j}^{\left(3\right)}=0,(j=16,17,18),$$

$$D_{12,1}^{\left(1\right)}=D_{10,1}^{\left(2\right)},D_{12,2}^{\left(1\right)}=D_{10,2}^{\left(2\right)},D_{12,3}^{\left(1\right)}=D_{10,3}^{\left(2\right)},D_{12,5}^{\left(1\right)}=D_{10,5}^{\left(2\right)},D_{12,8}^{\left(1\right)}=D_{10,6}^{\left(2\right)},$$

$$D_{12,9}^{\left(1\right)}=D_{10,7}^{\left(2\right)},D_{12,10}^{\left(1\right)}=D_{10,8}^{\left(2\right)},D_{12,12}^{\left(1\right)}=D_{10,10}^{\left(2\right)},D_{10,5}^{\left(1\right)}=D_{85}^{\left(2\right)},D_{10,12}^{\left(1\right)}=D_{8,10}^{\left(2\right)},$$

$$D_{18,1}^{\left(2\right)}=D_{12,1}^{\left(3\right)},D_{18,2}^{\left(2\right)}=D_{12,2}^{\left(3\right)},D_{18,3}^{\left(2\right)}=D_{12,3}^{\left(3\right)},$$

$$D_{18,6}^{\left(2\right)}=D_{12,4}^{\left(3\right)},D_{18,7}^{\left(2\right)}=D_{12,5}^{\left(3\right)},D_{18,8}^{\left(2\right)}=D_{12,6}^{\left(3\right)},$$

$$D_{18,16}^{\left(2\right)}=D_{12,10}^{\left(3\right)},D_{18,17}^{\left(2\right)}=D_{12,11}^{\left(3\right)},D_{18,18}^{\left(2\right)}=D_{12,12}^{\left(3\right)},$$

$$D_{8,16}^{\left(2\right)}=D_{6,10}^{\left(3\right)},D_{8,17}^{\left(2\right)}=D_{6,11}^{\left(3\right)},D_{8,18}^{\left(2\right)}=D_{6,12}^{\left(3\right)},$$

$$D_{10,1}^{\left(1\right)}+D_{61}^{\left(3\right)}=D_{81}^{\left(2\right)},D_{10,2}^{\left(1\right)}+D_{62}^{\left(3\right)}=D_{82}^{\left(2\right)},D_{10,3}^{\left(1\right)}+D_{63}^{\left(3\right)}=D_{83}^{\left(2\right)},$$

$$D_{10,8}^{\left(1\right)}+D_{64}^{\left(3\right)}=D_{86}^{\left(2\right)},D_{10,9}^{\left(1\right)}+D_{65}^{\left(3\right)}=D_{87}^{\left(2\right)},D_{10,10}^{\left(1\right)}+D_{66}^{\left(3\right)}=D_{88}^{\left(2\right)}.$$

$$r\left(\begin{array}{cccccc}{A}_1& C_1& B_1& 0& 0& 0\\ 0& {\left(B_1\right)}_{\varphi }& 0& {\left(A_2\right)}_{\varphi }& 0& 0\\ 0& 0& A_2& -C_2& B_2& 0\\ 0& 0& 0& {\left(B_2\right)}_{\varphi }& 0& {\left(A_3\right)}_{\varphi }\\ 0& 0& 0& 0& A_3& C_3\\ 0& 0& 0& 0& 0& {\left(B_3\right)}_{\varphi }\end{array}\right)=r\left(\begin{array}{ccc}{A}_1& B_1& 0\\ 0& A_2& B_2\\ 0& 0& A_3\end{array}\right)+r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\end{array}\right)$$

$$\iff r\left(\begin{array}{cccccc}{S}_{a_1}& D^{\left(1\right)}& S_{b_1}& 0& 0& 0\\ 0& {\left(S_{b_1}\right)}_{\varphi }& 0& {\left(S_{a_2}\right)}_{\varphi }& 0& 0\\ 0& 0& S_{a_2}& -D^{\left(2\right)}& S_{b_2}& 0\\ 0& 0& 0& {\left(S_{b_2}\right)}_{\varphi }& 0& {\left(S_{a_3}\right)}_{\varphi }\\ 0& 0& 0& 0& S_{a_3}& D^{\left(3\right)}\\ 0& 0& 0& 0& 0& {\left(S_{b_3}\right)}_{\varphi }\end{array}\right)$$

$$=r\left(\begin{array}{ccc}{S}_{a_1}& S_{b_1}& 0\\ 0& S_{a_2}& S_{b_2}\\ 0& 0& S_{a_3}\end{array}\right)+r\left(\begin{array}{ccc}{S}_{b_1}& 0& 0\\ S_{a_2}& S_{b_2}& 0\\ 0& S_{a_3}& S_{b_3}\end{array}\right)$$

$$\iff D_{i,14}^{\left(1\right)}=0,(i=8,\cdots ,14),D_{14,j}^{\left(1\right)}=0,(j=3,5,7,10,12),D_{i,7}^{\left(1\right)}=0,(i=8,\cdots ,10,12),$$

$$D_{i,20}^{\left(2\right)}=0,(i=1,2,3,5,6,7,8,10,16,17,18,20),D_{20,j}^{\left(2\right)}=0,(j=3,5,8,10,18,20),$$

$$D_{i,5}^{\left(2\right)}=0,(i=16,17,18),D_{i,10}^{\left(2\right)}=0,(i=16,17,18),D_{10,18}^{\left(2\right)}=0,$$

$$D_{i,18}^{\left(3\right)}=0,(i=4,5,6,10,11,12,16,17,18),$$

$$D_{i,3}^{\left(3\right)}=0,D_{i,6}^{\left(3\right)}=0,D_{i,12}^{\left(3\right)}=0,(i=16,17,18),$$

$$D_{12,3}^{\left(1\right)}=D_{10,3}^{\left(2\right)},D_{85}^{\left(1\right)}=D_{65}^{\left(2\right)},D_{95}^{\left(1\right)}=D_{75}^{\left(2\right)},D_{10,5}^{\left(1\right)}=D_{85}^{\left(2\right)},D_{12,5}^{\left(1\right)}=D_{10,5}^{\left(2\right)},$$

$$D_{12,10}^{\left(1\right)}=D_{10,8}^{\left(2\right)},D_{8,12}^{\left(1\right)}=D_{6,10}^{\left(2\right)},D_{9,12}^{\left(1\right)}=D_{7,10}^{\left(2\right)},D_{10,12}^{\left(1\right)}=D_{8,10}^{\left(2\right)},D_{12,12}^{\left(1\right)}=D_{10,10}^{\left(2\right)},$$

$$D_{6,18}^{\left(2\right)}=D_{4,12}^{\left(3\right)},D_{7,18}^{\left(2\right)}=D_{5,12}^{\left(3\right)},D_{8,18}^{\left(2\right)}=D_{6,12}^{\left(3\right)},$$

$$D_{16,3}^{\left(2\right)}=D_{10,3}^{\left(3\right)},D_{17,3}^{\left(2\right)}=D_{11,3}^{\left(3\right)},D_{18,3}^{\left(2\right)}=D_{12,3}^{\left(3\right)},$$

$$D_{16,8}^{\left(2\right)}=D_{10,6}^{\left(3\right)},D_{17,8}^{\left(2\right)}=D_{11,6}^{\left(3\right)},D_{18,8}^{\left(2\right)}=D_{12,6}^{\left(3\right)},$$

$$D_{16,18}^{\left(2\right)}=D_{10,12}^{\left(3\right)},D_{17,18}^{\left(2\right)}=D_{11,12}^{\left(3\right)},D_{18,18}^{\left(2\right)}=D_{12,12}^{\left(3\right)},$$

$$D_{83}^{\left(1\right)}+D_{43}^{\left(3\right)}=D_{63}^{\left(2\right)},D_{93}^{\left(1\right)}+D_{53}^{\left(3\right)}=D_{73}^{\left(2\right)},D_{10,3}^{\left(1\right)}+D_{63}^{\left(3\right)}=D_{83}^{\left(2\right)},$$

$$D_{8,10}^{\left(1\right)}+D_{46}^{\left(3\right)}=D_{68}^{\left(2\right)},D_{9,10}^{\left(1\right)}+D_{56}^{\left(3\right)}=D_{78}^{\left(2\right)},D_{10,10}^{\left(1\right)}+D_{66}^{\left(3\right)}=D_{88}^{\left(2\right)}.$$

$$r\left(\begin{array}{cccccc}{A}_2& C_2& B_2& 0& 0& 0\\ 0& {\left(B_2\right)}_{\varphi }& 0& {\left(A_3\right)}_{\varphi }& 0& 0\\ 0& 0& A_3& -C_3& B_3& 0\\ 0& 0& 0& {\left(B_3\right)}_{\varphi }& 0& {\left(A_4\right)}_{\varphi }\\ 0& 0& 0& 0& A_4& C_4\end{array}\right)=r\left(\begin{array}{ccc}{A}_2& B_2& 0\\ 0& A_3& B_3\\ 0& 0& A_4\end{array}\right)+r\left(\begin{array}{cc}{B}_2& 0\\ A_3& B_3\\ 0& A_4\end{array}\right)$$

$$\iff r\left(\begin{array}{cccccc}{S}_{a_2}& D^{\left(2\right)}& S_{b_2}& 0& 0& 0\\ 0& {\left(S_{b_2}\right)}_{\varphi }& 0& {\left(S_{a_3}\right)}_{\varphi }& 0& 0\\ 0& 0& S_{a_3}& -D^{\left(3\right)}& S_{b_3}& 0\\ 0& 0& 0& {\left(S_{b_3}\right)}_{\varphi }& 0& {\left(S_{a_4}\right)}_{\varphi }\\ 0& 0& 0& 0& S_{a_4}& D^{\left(4\right)}\end{array}\right)=r\left(\begin{array}{ccc}{S}_{a_2}& S_{b_2}& 0\\ 0& S_{a_3}& S_{b_3}\\ 0& 0& S_{a_4}\end{array}\right)+r\left(\begin{array}{cc}{S}_{b_2}& 0\\ S_{a_3}& S_{b_3}\\ 0& S_{a_4}\end{array}\right)$$

$$\iff D_{i,20}^{\left(2\right)}=0,(i=16,18,20),D_{20,j}^{\left(2\right)}=0,(j=1,3,5,6,8,10,11,13,15,16,18),$$

$$D_{16,5}^{\left(2\right)}=0,D_{18,5}^{\left(2\right)}=0,D_{16,10}^{\left(2\right)}=0,D_{18,10}^{\left(2\right)}=0,D_{16,15}^{\left(2\right)}=0,D_{18,15}^{\left(2\right)}=0,$$

$$D_{i,18}^{\left(3\right)}=0,(i=10,12,16,18),D_{18,j}^{\left(3\right)}=0,(i=1,3,4,6,7,9,10,12,16),$$

$$D_{12,16}^{\left(3\right)}=0,D_{16,3}^{\left(3\right)}=0,D_{16,6}^{\left(3\right)}=0,D_{16,9}^{\left(3\right)}=0,D_{16,12}^{\left(3\right)}=0,$$

$$D_{i8}^{\left(4\right)}=0,(i=4,6,8),D_{8j}^{\left(4\right)}=0,(j=1,2,3,4,6),$$

$$D_{16,3}^{\left(2\right)}=D_{10,3}^{\left(3\right)},D_{16,8}^{\left(2\right)}=D_{10,6}^{\left(3\right)},D_{16,13}^{\left(2\right)}=D_{10,9}^{\left(3\right)},$$

$$D_{16,18}^{\left(2\right)}=D_{10,12}^{\left(3\right)},D_{18,1}^{\left(2\right)}=D_{12,1}^{\left(3\right)},D_{18,3}^{\left(2\right)}=D_{12,3}^{\left(3\right)},$$

$$D_{18,6}^{\left(2\right)}=D_{12,4}^{\left(3\right)},D_{18,8}^{\left(2\right)}=D_{12,6}^{\left(3\right)},D_{18,11}^{\left(2\right)}=D_{12,7}^{\left(3\right)},$$

$$D_{18,13}^{\left(2\right)}=D_{12,9}^{\left(3\right)},D_{18,16}^{\left(2\right)}=D_{12,10}^{\left(3\right)},D_{18,18}^{\left(2\right)}=D_{12,12}^{\left(3\right)},$$

$$D_{10,16}^{\left(3\right)}=D_{46}^{\left(4\right)},D_{16,1}^{\left(3\right)}=D_{61}^{\left(4\right)},D_{16,4}^{\left(3\right)}=D_{62}^{\left(4\right)},$$

$$D_{16,7}^{\left(3\right)}=D_{63}^{\left(4\right)},D_{16,10}^{\left(3\right)}=D_{64}^{\left(4\right)},D_{16,16}^{\left(3\right)}=D_{66}^{\left(4\right)},$$

$$D_{16,1}^{\left(2\right)}+D_{41}^{\left(4\right)}=D_{10,1}^{\left(3\right)},D_{16,6}^{\left(2\right)}+D_{42}^{\left(4\right)}=D_{10,4}^{\left(3\right)},$$

$$D_{16,11}^{\left(2\right)}+D_{43}^{\left(4\right)}=D_{10,7}^{\left(3\right)},D_{16,16}^{\left(2\right)}+D_{44}^{\left(4\right)}=D_{10,10}^{\left(3\right)}.$$

$$r\left(\begin{array}{cccccccc}{A}_1& C_1& B_1& 0& 0& 0& 0& 0\\ 0& {\left(B_1\right)}_{\varphi }& 0& {\left(A_2\right)}_{\varphi }& 0& 0& 0& 0\\ 0& 0& A_2& -C_2& B_2& 0& 0& 0\\ 0& 0& 0& {\left(B_2\right)}_{\varphi }& 0& {\left(A_3\right)}_{\varphi }& 0& 0\\ 0& 0& 0& 0& A_3& C_3& B_3& 0\\ 0& 0& 0& 0& 0& {\left(B_3\right)}_{\varphi }& 0& {\left(A_4\right)}_{\varphi }\\ 0& 0& 0& 0& 0& 0& A_4& -C_4\end{array}\right)=r\left(\begin{array}{cccc}{A}_1& B_1& 0& 0\\ 0& A_2& B_2& 0\\ 0& 0& A_3& B_3\\ 0& 0& 0& A_4\end{array}\right)+r\left(\begin{array}{ccc}{B}_1& 0& 0\\ A_2& B_2& 0\\ 0& A_3& B_3\\ 0& 0& A_4\end{array}\right)$$

$$\iff r\left(\begin{array}{cccccccc}{S}_{a_1}& D^{\left(1\right)}& S_{b_1}& 0& 0& 0& 0& 0\\ 0& {\left(S_{b_1}\right)}_{\varphi }& 0& {\left(S_{a_2}\right)}_{\varphi }& 0& 0& 0& 0\\ 0& 0& S_{a_2}& -D^{\left(2\right)}& S_{b_2}& 0& 0& 0\\ 0& 0& 0& {\left(S_{b_2}\right)}_{\varphi }& 0& {\left(S_{a_3}\right)}_{\varphi }& 0& 0\\ 0& 0& 0& 0& S_{a_3}& D^{\left(3\right)}& S_{b_3}& 0\\ 0& 0& 0& 0& 0& {\left(S_{b_3}\right)}_{\varphi }& 0& {\left(S_{a_4}\right)}_{\varphi }\\ 0& 0& 0& 0& 0& 0& S_{a_4}& -D^{\left(4\right)}\end{array}\right)$$

$$=r\left(\begin{array}{cccc}{S}_{a_1}& S_{b_1}& 0& 0\\ 0& S_{a_2}& S_{b_2}& 0\\ 0& 0& S_{a_3}& S_{b_3}\\ 0& 0& 0& S_{a_4}\end{array}\right)+r\left(\begin{array}{ccc}{S}_{b_1}& 0& 0\\ S_{a_2}& S_{b_2}& 0\\ 0& S_{a_3}& S_{b_3}\\ 0& 0& S_{a_4}\end{array}\right)$$

$$\iff D_{14,j}^{\left(1\right)}=0,(j=1,3,5,7,8,10,12,14),D_{87}^{\left(1\right)}=0,D_{8,14}^{\left(1\right)}=0,D_{10,7}^{\left(1\right)}=0,$$

$$D_{10,14}^{\left(1\right)}=0,D_{12,7}^{\left(1\right)}=0,D_{12,14}^{\left(1\right)}=0,$$

$$D_{20,j}^{\left(2\right)}=0,(j=1,3,5,6,8,10,16,18,20),D_{10,16}^{\left(2\right)}=0,D_{10,18}^{\left(2\right)}=0,D_{10,20}^{\left(2\right)}=0,$$

$$D_{6,20}^{\left(2\right)}=0,D_{8,20}^{\left(2\right)}=0,D_{16,5}^{\left(2\right)}=0,$$

$$D_{16,10}^{\left(2\right)}=0,D_{16,20}^{\left(2\right)}=0,D_{18,5}^{\left(2\right)}=0,D_{18,10}^{\left(2\right)}=0,D_{18,20}^{\left(2\right)}=0,$$

$$D_{i,18}^{\left(3\right)}=0,(i=4,6,10,12,16,18),D_{18,j}^{\left(3\right)}=0,(j=1,3,4,6,10,12,16,18),$$

$$D_{16,3}^{\left(3\right)}=0,D_{16,6}^{\left(3\right)}=0,D_{16,12}^{\left(3\right)}=0,D_{6,16}^{\left(3\right)}=0,D_{12,16}^{\left(3\right)}=0,$$

$$D_{i8}^{\left(4\right)}=0,(i=2,4,6,8),D_{8j}^{\left(4\right)}=0,(j=1,2,4,6),$$

$$D_{85}^{\left(1\right)}=D_{65}^{\left(2\right)},D_{8,12}^{\left(1\right)}=D_{6,10}^{\left(2\right)},D_{10,5}^{\left(1\right)}=D_{85}^{\left(2\right)},D_{10,12}^{\left(1\right)}=D_{8,10}^{\left(2\right)},$$

$$D_{12,1}^{\left(1\right)}=D_{10,1}^{\left(2\right)},D_{12,3}^{\left(1\right)}=D_{10,3}^{\left(2\right)},D_{12,5}^{\left(1\right)}=D_{10,5}^{\left(2\right)},$$

$$D_{12,8}^{\left(1\right)}=D_{10,6}^{\left(2\right)},D_{12,10}^{\left(1\right)}=D_{10,8}^{\left(2\right)},D_{12,12}^{\left(1\right)}=D_{10,10}^{\left(2\right)},$$

$$D_{6,18}^{\left(2\right)}=D_{4,12}^{\left(3\right)},D_{8,16}^{\left(2\right)}=D_{6,10}^{\left(3\right)},D_{8,18}^{\left(2\right)}=D_{6,12}^{\left(3\right)},$$

$$D_{16,3}^{\left(2\right)}=D_{10,3}^{\left(3\right)},D_{16,8}^{\left(2\right)}=D_{10,6}^{\left(3\right)},D_{16,18}^{\left(2\right)}=D_{10,12}^{\left(3\right)},$$

$$D_{18,1}^{\left(2\right)}=D_{12,1}^{\left(3\right)},D_{18,3}^{\left(2\right)}=D_{12,3}^{\left(3\right)},D_{18,6}^{\left(2\right)}=D_{12,4}^{\left(3\right)},$$

$$D_{18,8}^{\left(2\right)}=D_{12,6}^{\left(3\right)},D_{18,16}^{\left(2\right)}=D_{12,10}^{\left(3\right)},D_{18,18}^{\left(2\right)}=D_{12,12}^{\left(3\right)},$$

$$D_{16,1}^{\left(3\right)}=D_{61}^{\left(4\right)},D_{16,4}^{\left(3\right)}=D_{62}^{\left(4\right)},D_{16,10}^{\left(3\right)}=D_{64}^{\left(4\right)},$$

$$D_{16,16}^{\left(3\right)}=D_{66}^{\left(4\right)},D_{4,16}^{\left(3\right)}=D_{26}^{\left(4\right)},D_{10,16}^{\left(3\right)}=D_{46}^{\left(4\right)},$$

$$D_{10,1}^{\left(1\right)}+D_{61}^{\left(3\right)}=D_{81}^{\left(2\right)},D_{10,3}^{\left(1\right)}+D_{63}^{\left(3\right)}=D_{83}^{\left(2\right)},$$

$$D_{10,8}^{\left(1\right)}+D_{64}^{\left(3\right)}=D_{86}^{\left(2\right)},D_{10,10}^{\left(1\right)}+D_{66}^{\left(3\right)}=D_{88}^{\left(2\right)},$$

$$D_{83}^{\left(1\right)}+D_{43}^{\left(3\right)}=D_{63}^{\left(2\right)},D_{8,10}^{\left(1\right)}+D_{46}^{\left(3\right)}=D_{68}^{\left(2\right)},$$

$$D_{16,1}^{\left(2\right)}+D_{41}^{\left(4\right)}=D_{10,1}^{\left(3\right)},D_{16,6}^{\left(2\right)}+D_{42}^{\left(4\right)}=D_{10,4}^{\left(3\right)},$$

$$D_{16,16}^{\left(2\right)}+D_{44}^{\left(4\right)}=D_{10,10}^{\left(3\right)},D_{6,16}^{\left(2\right)}+D_{24}^{\left(4\right)}=D_{4,10}^{\left(3\right)},$$

$$D_{81}^{\left(1\right)}+D_{41}^{\left(3\right)}=D_{61}^{\left(2\right)}+D_{21}^{\left(4\right)},D_{88}^{\left(1\right)}+D_{44}^{\left(3\right)}=D_{66}^{\left(2\right)}+D_{22}^{\left(4\right)}.$$

□

In Theorem 2, let $A_3$, $B_3$, $A_4$, and $C_3$, $C_4$ vanish, then we can obtain the necessary and sufficient conditions for the existence of a $\varphi$-skew-Hermitian solution in the following equation:

Corollary 1. 

Let $C_1=-{\left(C_1\right)}_{\varphi }\in {\mathbb{H}}^{p\times p}$, $A_1\in {\mathbb{H}}^{p\times l}$, $B_1\in {\mathbb{H}}^{p\times n}$, $A_2\in {\mathbb{H}}^{q\times n}$, $B_2\in {\mathbb{H}}^{q\times k}$, and $C_2=-{\left(C_2\right)}_{\varphi }\in {\mathbb{H}}^{q\times q}$ are given. The system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{A}_1{X}_1{\left(A_1\right)}_{\varphi }+B_1{X}_2{\left(B_1\right)}_{\varphi }=C_1,\\ A_2{X}_2{\left(A_2\right)}_{\varphi }+B_2{X}_3{\left(B_2\right)}_{\varphi }=C_2\end{array}\right.X_i=-{\left(X_i\right)}_{\varphi },\end{array}$$

(33)

has a ϕ-skew-Hermitian solution $(X_1,X_2,X_3)\in {\mathbb{H}}^{l\times l}\times {\mathbb{H}}^{n\times n}\times {\mathbb{H}}^{k\times k}$ if and only if the ranks satisfy:

$$r(A_1,C_1,B_1)=r(A_1,B_1),r(A_2,C_2,B_2)=r(A_2,B_2),$$

$$r\left(\begin{array}{cc}{A}_1& C_1\\ 0& {\left(B_1\right)}_{\varphi }\end{array}\right)=r\left(A_1\right)+r\left(B_1\right),r\left(\begin{array}{cc}{A}_2& C_2\\ 0& {\left(B_2\right)}_{\varphi }\end{array}\right)=r\left(A_2\right)+r\left(B_2\right),$$

$$r\left(\begin{array}{ccccc}0& {\left(B_1\right)}_{\varphi }& {\left(A_2\right)}_{\varphi }& 0& 0\\ B_1& C_1& 0& A_1& 0\\ A_2& 0& -\left(C_2\right)& 0& B_2\end{array}\right)=r\left(\begin{array}{ccc}{A}_1& B_1& 0\\ 0& A_2& B_2\end{array}\right)+r\left(\begin{array}{c}{B}_1\\ A_2\end{array}\right),$$

$$r\left(\begin{array}{cccc}0& {\left(B_1\right)}_{\varphi }& {\left(A_2\right)}_{\varphi }& 0\\ B_1& C_1& 0& A_1\\ A_2& 0& -\left(C_2\right)& 0\\ 0& 0& {\left(B_2\right)}_{\varphi }& 0\end{array}\right)=r\left(\begin{array}{cc}{A}_1& B_1\\ 0& A_2\end{array}\right)+r\left(\begin{array}{cc}{B}_1& 0\\ A_2& B_2\end{array}\right).$$

Remark 1. 

Corollary 1 is the main result of [31].

4. Conclusions

We investigated some necessary and sufficient conditions for the existence of a $\varphi$-skew-Hermitian solution to the system (1) by using a simultaneous decomposition for a set of quaternion matrices. Some of the known results can be considered special cases in this paper.