Assessing the safe haven properties of oil in African stock markets amid the COVID-19 pandemic: a quantile regression analysis

Emmanuel Assifuah-Nunoo; Peterson Owusu Junior; Anokye Mohammed Adam; Ahmed Bossman; Emmanuel Assifuah-Nunoo; Peterson Owusu Junior; Anokye Mohammed Adam; Ahmed Bossman

doi:10.3934/QFE.2022011

Quantitative Finance and Economics

2022, Volume 6, Issue 2: 244-269. doi: 10.3934/QFE.2022011

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Research article

Assessing the safe haven properties of oil in African stock markets amid the COVID-19 pandemic: a quantile regression analysis

Department of Finance, School of Business, University of Cape Coast, Cape Coast, Ghana

Received: 31 March 2022 Revised: 23 May 2022 Accepted: 24 May 2022 Published: 27 May 2022
JEL Codes: G01, G11, G15, F21, Q43, C22

Using the quantile regression approach to reveal the conditional relationships, the study re-examined the oil-stock co-movement in the context of oil-exporting countries in Africa. The data employed include daily OPEC basket price for crude oil and daily data on stock market indices for six major stock markets of oil-exporting economies in Africa—Egypt, Ghana, Morocco, Nigeria, South Africa, and Tunisia, from 02 January 2020 to 06 May 2021. We found that crude oil cannot act as safe haven instrument for stock markets in oil-exporting African countries. Notably, the oil-stock co-movement is consistent and more intense at the lower tails only. Investors are encouraged to employ oil as a diversification instrument rather than as a safe haven asset, based on market conditions. Regulators should devise strategies to strengthen the market for crude oil to lessen adverse volatilities during the COVID-19 pandemic by way of mitigating downward returns in African stock markets. The findings of the study offer more interesting economic insights to all classes of investors as well as policymakers in oil-exporting economies in Africa.

Keywords:

Citation: Emmanuel Assifuah-Nunoo, Peterson Owusu Junior, Anokye Mohammed Adam, Ahmed Bossman. Assessing the safe haven properties of oil in African stock markets amid the COVID-19 pandemic: a quantile regression analysis[J]. Quantitative Finance and Economics, 2022, 6(2): 244-269. doi: 10.3934/QFE.2022011

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Abstract

1. Introduction

In this paper, we establish the following four symmetric quaternion matrix systems:

$\begin{align} \begin{aligned} \left\{\begin{array}{c} A_{11}X_1 = B_{11}, C_{11}X_1D_{11} = E_{11},\\ X_2A_{22} = B_{22}, C_{22}X_2D_{22} = E_{22},\\ F_{11}X_1H_{11}+X_2F_{22} = G_{11}, \end{array} \right. \end{aligned} \end{align}$

(1.1)

$\begin{align} \begin{aligned} \left\{\begin{array}{c} A_{11}X_1 = B_{11}, C_{11}X_1D_{11} = E_{11},\\ X_2A_{22} = B_{22}, C_{22}X_2D_{22} = E_{22},\\ F_{11}X_1+H_{11}X_2F_{22} = G_{11}, \end{array} \right. \end{aligned} \end{align}$

(1.2)

$\begin{align} \begin{aligned} \left\{\begin{array}{c} A_{11}X_1 = B_{11}, C_{11}X_1D_{11} = E_{11},\\ A_{22}X_2 = B_{22}, C_{22}X_2D_{22} = E_{22},\\ F_{11}X_1+H_{11}X_2F_{22} = G_{11}, \end{array} \right. \end{aligned} \end{align}$

(1.3)

$\begin{align} \begin{aligned} \left\{\begin{array}{c} A_{11}X_1 = B_{11}, C_{11}X_1D_{11} = E_{11},\\ A_{22}X_2 = B_{22}, C_{22}X_2D_{22} = E_{22},\\ F_{11}X_1+X_2F_{22} = G_{11}, \end{array} \right. \end{aligned} \end{align}$

(1.4)

where $A_{ii}, \ B_{ii}, \ C_{ii}, \ D_{ii}, \ E_{ii}, \ F_{ii}(i = \overline{1, 2})$ , $H_{11}$ , and $G_{11}$ are known matrices, while $X_i (i = \overline{1, 2})$ are unknown.

In this paper, $\mathbb{R}$ and $\mathbb{H}^{m\times n}$ denote the real number field and the set of all quaternion matrices of order $m\times n$ , respectively.

$\begin{align*} &\mathbb{H} = \{v_{0}+v_{1}\mathbf{i}+v_{2}\mathbf{j}+v_{3}\mathbf{k} | \mathbf{i}^{2} = \mathbf{j}^{2} = \mathbf{k}^{2} = \mathbf{ijk} = -1, v_{0}, v_{1}, v_{2}, v_{3}\in \mathbb{R}\}. \end{align*}$

Moreover, $r(A)$ , $0$ and $I$ represent the rank of matrix $A$ , the zero matrix of suitable size, and the identity matrix of suitable size, respectively. The conjugate transpose of $A$ is $A^{\ast}$ . For any matrix $A$ , if there exists a unique solution $X$ such that

$\begin{align*} AXA = A, XAX = X, (AX)^{\ast} = AX, (XA)^{\ast} = XA, \end{align*}$

then $X$ is called the Moore-Penrose ( $M-P$ ) inverse. It should be noted that $A^{\dagger}$ is used to represent the $M-P$ inverse of $A$ . Additionally, $L_{A} = I-A^{\dagger}A$ and $R_{A} = I-AA^{\dagger}$ denote two projectors toward $A$ .

$\mathbb{H}$ is known to be an associative noncommutative division algebra over $\mathbb{R}$ with extensive applications in computer science, orbital mechanics, signal and color image processing, control theory, and so on (see ^[1,2,3,4]).

Matrix equations, significant in the domains of descriptor systems control theory ^[5], nerve networks ^[6], back feed ^[7], and graph theory ^[8], are one of the key research topics in mathematics.

The study of matrix equations in $\mathbb{H}$ has garnered the attention of various researchers; consequently they have been analyzed by many studies (see, e.g., ^[9,10,11,12]). Among these the system of symmetric matrix equations is a crucial research object. For instance, Mahmoud and Wang ^[13] established some necessary and sufficient conditions for the three symmetric matrix systems in terms of $M-P$ inverses and rank equalities:

$\begin{equation} \begin{aligned} \left\{\begin{array}{c} A_{1}V = C_1,\ VB_1 = C_2,\\ A_3X+YB_3 = C_3,\\ A_2Y+ZB_2+A_5VB_5 = C_5,\\ A_4W+ZB_4 = C_4,\\ \end{array} \right. \left\{\begin{array}{c} A_{1}V = C_1,\ VB_1 = C_2,\\ A_3X+YB_3 = C_3,\\ A_2Z+YB_2+A_5VB_5 = C_5,\\ A_4Z+WB_4 = C_4,\\ \end{array} \right. \left\{\begin{array}{c} A_{1}V = C_1,\ VB_1 = C_2,\\ A_3X+YB_3 = C_3,\\ A_2Y+ZB_2+A_5VB_5 = C_5,\\ A_4Z+WB_4 = C_4. \end{array} \right. \end{aligned} \end{equation}$

(1.5)

Wang and He ^[14] established the sufficient and necessary conditions for the existence of solutions to the following three symmetric coupled matrix equations and the expressions for their general solutions:

$\begin{equation} \begin{aligned} \left\{\begin{array}{c} A_{1}X+YB_1 = C_1,\\ A_2Y+ZB_2 = C_2,\\ A_3W+ZB_3 = C_3,\\ \end{array} \right. \left\{\begin{array}{c} A_{1}X+YB_1 = C_1,\\ A_2Z+YB_2 = C_2,\\ A_3Z+WB_3 = C_3,\\ \end{array} \right. \left\{\begin{array}{c} A_{1}X+YB_1 = C_1,\\ A_2Y+ZB_2 = C_2,\\ A_3Z+WB_3 = C_3. \\ \end{array} \right. \end{aligned} \end{equation}$

(1.6)

It is noteworthy that the following matrix equation plays an important role in the analysis of the solvability conditions of systems (1.1)–(1.4):

$\begin{equation} \begin{aligned} A_1U+VB_1+A_2XB_2+A_3YB_3+A_4ZB_4 = B. \end{aligned} \end{equation}$

(1.7)

Liu et al. ^[15] derived some necessary and sufficient conditions to solve the quaternion matrix equation (1.7) using the ranks of coefficient matrices and $M-P$ inverses. Wang et al. ^[16] derived the following quaternion equations after obtaining some solvability conditions for the quaternion equation presented in Eq (1.8) in terms of $M-P$ inverses:

$\begin{equation} \begin{aligned} \left\{\begin{array}{c} A_{11}X_1 = B_{11}, C_{11}X_1D_{11} = E_{11},\\ X_2A_{22} = B_{22}, C_{22}X_2D_{22} = E_{22},\\ F_{11}X_1+X_2F_{22} = G_{11}. \end{array} \right. \end{aligned} \end{equation}$

(1.8)

To our knowledge, so far, there has been little information on the solvability conditions and an expression of the general solution to systems (1.1)–(1.4).

In mathematical research and applications, the concept of $\eta$ -Hermitian matrices has gained significant attention ^[17]. An $\eta$ -Hermitian matrix, for $\eta \in \{\mathbf{i}, \mathbf{j}, \mathbf{k}\}$ , is defined as a matrix $A$ such that $A = A^{\eta^{*}}$ , where $A^{\eta^{*}} = -\eta A^{*}\eta$ . These matrices have found applications in various fields including linear modeling and the statistics of random signals ^[1,17]. As an application of (1.1), this paper establishes some necessary and sufficient conditions for the following matrix equation:

$\begin{equation} \begin{aligned} \left\{\begin{array}{c} A_{11}X_1 = B_{11}, C_{11}X_1C_{11}^{\eta^{*}} = E_{11},\\ F_{11}X_1F_{11}^{\eta^{*}}+(F_{22}X_1)^{\eta^{*}} = G_{11} \end{array} \right. \end{aligned} \end{equation}$

(1.9)

to be solvable.

Motivated by the study of Systems (1.8), symmetric matrix equations, $\eta$ -Hermitian matrices, and the widespread use of matrix equations and quaternions as well as the need for their theoretical advancements, we examine the solvability conditions of the quaternion systems presented in systems (1.1)–(1.4) by utilizing the rank equalities and the $M-P$ inverses of coefficient matrices. We then obtain the general solutions for the solvable quaternion equations in systems (1.1)–(1.4). As an application of (1.1), we utilize the $M-P$ inverse and the rank equality of matrices to investigate the necessary and sufficient conditions for the solvability of quaternion matrix equations involving $\eta$ -Hermicity matrices. It is evident that System (1.8) is a specific instance of System (1.1).

The remainder of this article is structured as follows. Section 2 outlines the basics. Section 3 examines some solvability conditions of the quaternion equation presented in System (1.1) using the $M-P$ inverses and rank equalities of the matrices, and derives the solution of System (1.1). Section 4 establishes some solvability conditions for matrix systems (1.2)–(1.4) to be solvable. Section 5 investigates some necessary and sufficient conditions for matrix equation (1.9) to have common solutions. Section 6 concludes the paper.

2. Preliminaries

Marsaglia and Styan ^[18] presented the following rank equality lemma over the complex field, which can be generalized to $\mathbb{H}$ .

Lemma 2.1. ^[18] Let $A \in \mathbb{H}^{m\times n}$ , $B \in \mathbb{H}^{m\times k}$ , $C \in \mathbb{H}^{l\times n}$ , $D \in \mathbb{H}^{j\times k},$ and $E \in \mathbb{H}^{l\times i}$ be given. Then, the following rank equality holds:

$\ r\begin{pmatrix} A&BL_{D}\\ R_{E}C&0 \end{pmatrix} = r\begin{pmatrix} A&B&0\\ C&0&E\\ 0&D&0 \end{pmatrix}-r\begin{pmatrix} D \end{pmatrix}-r\begin{pmatrix} E \end{pmatrix}.$

Lemma 2.2. ^[19] Let $A \in \mathbb{H}^{m\times n}$ be given. Then,

$\begin{align*} &(1)\quad (A^{\eta})^{\dagger} = (A^{\dagger})^{\eta}, (A^{\eta^{*}})^{\dagger} = (A^{\dagger})^{\eta^{*}};\\ & (2)\quad r(A) = r(A^{\eta^{*}}) = r(A^{\eta});\\ & (3)\quad (L_A)^{\eta^{*}} = -\eta(L_A)\eta = (L_A)^{\eta} = L_{A^{\eta^{*}}} = R_{A^{\eta^{*}}},\\ & (4)\quad (R_A)^{\eta^{*}} = -\eta(R_A)\eta = (R_A)^{\eta} = R_{A^{\eta^{*}}} = L_{A^{\eta^{*}}};\\ & (5)\quad (AA^{\dagger})^{\eta^{*}} = (A^{\dagger})^{\eta^{*}}A^{\eta^{*}} = (A^{\dagger}A)^{\eta} = A^{\eta}(A^{\dagger})^{\eta};\\ & (6)\quad (A^{\dagger}A)^{\eta^{*}} = A^{\eta^{*}}(A^{\dagger})^{\eta^{*}} = (AA^{\dagger})^{\eta} = (A^{\dagger})^{\eta}A^{\eta}. \end{align*}$

Lemma 2.3. ^[20] Let $A_{1}$ and $A_{2}$ be given quaternion matrices with adequate shapes. Then, the equation $A_{1}X = A_{2}$ is solvable if, and only if, $A_{2} = A_{1}A_{1}^{\dagger}A_{2}$ . In this case, the general solution to this equation can be expressed as

$X = A_{1}^{\dagger}A_{2}+L_{A_{1}}U_{1},$

where $U_{1}$ is any matrix with appropriate size.

Lemma 2.4. ^[20] Let $A_{1}$ and $A_{2}$ be given quaternion matrices with adequate shapes. Then, the equation $XA_{1} = A_{2}$ is solvable if, and only if, $A_{2} = A_{2}A_{1}^{\dagger}A_{1}$ . In this case, the general solution to this equation can be expressed as

$X = A_{2}A_{1}^{\dagger}+U_{1}R_{A_{1}},$

where $U_{1}$ is any matrix with appropriate size.

Lemma 2.5. ^[21] Let $A, B,$ and $C$ be known quaternion matrices with appropriate sizes. Then, the matrix equation

$\begin{align*} AXB = C \end{align*}$

is consistent if, and only if,

$\begin{align*} R_{A}C = 0,CL_{B} = 0. \end{align*}$

In this case, the general solution to this equation can be expressed as

$\begin{align*} X = A^{\dagger}CB^{\dagger}+L_{A}U+VR_{B}, \end{align*}$

where $U$ and $V$ are any quaternion matrices with appropriate sizes.

Lemma 2.6. ^[15] Let $C_{i}, D_{i}$ , and $Z \; (i = \overline{1, 4})$ be known quaternion matrices with appropriate sizes.

$\begin{equation} C_{1}X_{1}+X_{2}D_{1}+C_{2}Y_{1}D_{2}+C_{3}Y_{2}D_{3}+C_{4}Y_{3}D_{4} = Z. \end{equation}$

(2.1)

Denote

$\begin{align*} &R_{C_{1}}C_{2} = C_{12},R_{C_{1}}C_{3} = C_{13},R_{C_{1}}C_{4} = C_{14},D_{2}L_{D_{1}} = D_{21},D_{31}L_{D_{21}} = N_{32},D_{3}L_{D_{1}} = D_{31},\\ &D_{4}L_{D_{1}} = D_{41},R_{C_{12}}C_{13} = M_{23},S_{12} = C_{13}L_{M_{23}},R_{C_{1}}ZL_{D_{1}} = T_{1},C_{32} = R_{M_{23}}R_{C_{12}},A_{1} = C_{32}C_{14},\\ & A_{2} = R_{C_{12}}C_{14},A_{3} = R_{C_{13}}C_{14},A_{4} = C_{14},D_{13} = L_{D_{21}}L_{N_{32}},B_{1} = D_{41},B_{2} = D_{41}L_{D_{31}},\\ &B_{3} = D_{41}L_{D_{21}},B_{4} = D_{41}D_{13},E_{1} = C_{32}T_{1},E_{2} = R_{C_{12}}T_{1}L_{D_{31}},E_{3} = R_{C_{13}}T_{1}L_{D_{21}},E_{4} = T_{1}D_{13}, \\ &A_{24} = (L_{A_{2}},L_{A_{4}}),B_{13} = \begin{pmatrix} R_{B_{1}}\\R_{B_{3}} \end{pmatrix},A_{11} = L_{A_{1}},B_{22} = R_{B_{2}},A_{33} = L_{A_{3}},B_{44} = R_{B_{4}},E_{11} = R_{A_{24}}A_{11},\\ &E_{22} = R_{A_{24}}A_{33},E_{33} = B_{22}L_{B_{13}},E_{44} = B_{44}L_{B_{13}},N = R_{E_{11}}E_{22},M = E_{44}L_{E_{33}},K = K_{2}-K_{1},\\ &E = R_{A_{24}}KL_{B_{13}},S = E_{22}L_{N},K_{11} = A_{2}L_{A_{1}},G_{1} = E_{2}-A_{2}A_{1}^{\dagger}E_{1}B_{1}^{\dagger}B_{2},K_{22} = A_{4}L_{A_{3}},\\ &G_{2} = E_{4}-A_{4}A_{3}^{\dagger}E_{3}B_{3}^{\dagger}B_{4},K_{1} = A_{1}^{\dagger}E_{1}B_{1}^{\dagger}+L_{A_{1}}A_{2}^{\dagger}E_{2}B_{2}^{\dagger},K_{2} = A_{3}^{\dagger}E_{3}B_{3}^{\dagger}+L_{A_{3}}A_{4}^{\dagger}E_{4}B_{4}^{\dagger}. \end{align*}$

Then, the following statements are equivalent:

$\mathrm{(1)}$ Equation (2.1) is consistent.

$\mathrm{(2)}$

$\begin{align*} R_{A_{i}}E_{i} = 0,E_{i}L_{B_{i}} = 0(i = \overline{1,4}),R_{E_{11}}EL_{E_{44}} = 0. \end{align*}$

$\mathrm{(3)}$

$\begin{align*} \quad \quad \quad&r\begin{pmatrix} Z&C_{2}&C_{3}&C_{4}&C_{1}\\D_{1}&0&0&0&0 \end{pmatrix} = r(D_{1})+r(C_{2},C_{3},C_{4},C_{1}),\\ &r\begin{pmatrix} Z&C_{2}&C_{4}&C_{1}\\D_{3}&0&0&0\\D_{1}&0&0&0 \end{pmatrix} = r(C_{2},C_{4},C_{1})+r\begin{pmatrix} D_{3}\\D_{1} \end{pmatrix},\\ &r\begin{pmatrix} Z&C_{3}&C_{4}&C_{1}\\D_{2}&0&0&0\\D_{1}&0&0&0 \end{pmatrix} = r(C_{3},C_{4},C_{1})+r\begin{pmatrix} D_{2}\\D_{1} \end{pmatrix},\\ &r\begin{pmatrix} Z&C_{4}&C_{1}\\D_{2}&0&0\\D_{3}&0&0\\D_{1}&0&0 \end{pmatrix} = r\begin{pmatrix} D_{2}\\D_{3}\\D_{1} \end{pmatrix}+r(C_{4},C_{1}),\\ & r\begin{pmatrix} Z&C_{2}&C_{3}&C_{1}\\D_{4}&0&0&0\\D_{1}&0&0&0 \end{pmatrix} = r(C_{2},C_{3},C_{1})+r\begin{pmatrix} D_{4}\\D_{1} \end{pmatrix},\\ & r\begin{pmatrix} Z&C_{2}&C_{1}\\D_{3}&0&0\\D_{4}&0&0\\D_{1}&0&0 \end{pmatrix} = r\begin{pmatrix} D_{3}\\D_{4}\\D_{1} \end{pmatrix}+r(C_{2},C_{1}), \\ & r\begin{pmatrix} Z&C_{3}&C_{1}\\D_{2}&0&0\\D_{4}&0&0\\D_{1}&0&0 \end{pmatrix} = r\begin{pmatrix} D_{2}\\D_{4}\\D_{1} \end{pmatrix}+r(C_{3},C_{1}),\\\\ & r\begin{pmatrix} Z&C_{1}\\D_{2}&0\\D_{3}&0\\D_{4}&0\\D_{1}&0 \end{pmatrix} = r\begin{pmatrix} D_{2}\\D_{3}\\D_{4}\\D_{1} \end{pmatrix}+r(C_{1}),\\ &r\begin{pmatrix} Z&C_{2}&C_{1}&0&0&0&C_{4}\\D_{3}&0&0&0&0&0&0\\D_{1}&0&0&0&0&0&0\\0&0&0&-Z&C_{3}&C_{1}&C_{4}\\0&0&0&D_{2}&0&0&0\\0&0&0&D_{1}&0&0&0\\D_{4}&0&0&D_{4}&0&0&0 \end{pmatrix} = r\begin{pmatrix} D_{3}&0\\D_{1}&0\\0&D_{2}\\0&D_{1}\\D_{4}&D_{4} \end{pmatrix}+r\begin{pmatrix} C_{2}&C_{1}&0&0&C_{4}\\0&0&C_{3}&C_{1}&C_{4} \end{pmatrix}. \end{align*}$

Under these conditions, the general solution to the matrix equation (2.1) is

$\begin{align*} &X_{1} = C_{1}^{\dagger}(Z-C_{2}Y_{1}D_{2}-C_{3}Y_{2}D_{3}-C_{4}Y_{3}D_{4})-C_{1}^{\dagger}U_{1}D_{1}+L_{C_{1}}U_{2},\\ &X_{2} = R_{C_{1}}(Z-C_{2}Y_{1}D_{2}-C_{3}Y_{2}D_{3}-C_{4}Y_{3}D_{4})D_{1}^{\dagger}+C_{1}C_{1}^{\dagger}U_{1}+U_{3}R_{D_{1}},\\ &Y_{1} = C_{12}^{\dagger}TD_{21}^{\dagger}-C_{12}^{\dagger}C_{13}M_{23}^{\dagger}TD_{21}^{\dagger}-C_{12}^{\dagger}S_{12}C_{13}^{\dagger}TN_{32}^{\dagger}D_{31}D_{21}^{\dagger}\\ &\quad\quad-C_{12}^{\dagger}S_{12}U_{4}R_{N_{32}}D_{31}D_{21}^{\dagger}+L_{C_{12}}U_{5}+U_{6}R_{D_{21}},\\ &Y_{2} = M_{23}^{\dagger}TD_{31}^{\dagger}+S_{12}^{\dagger}S_{12}C_{13}^{\dagger}TN_{32}^{\dagger}+L_{M_{23}}L_{S_{12}}U_{7}+U_{8}R_{D_{31}}+L_{M_{23}}U_{4}R_{N_{32}},\\ &Y_{3} = K_{1}+L_{A_{2}}V_{1}+V_{2}R_{B_{1}}+L_{A_{1}}V_{3}R_{B_{2}},\ or\ Y_{3} = K_{2}-L_{A_{4}}W_{1}-W_{2}R_{B_{3}}-L_{A_{3}}W_{3}R_{B_{4}}, \end{align*}$

where $T = T_{1}-C_{4}Y_{3}D_{4}, U_{i}(i = \overline{1, 8})$ are arbitrary matrices with appropriate sizes over $\mathbb{H}$ ,

$\begin{align*} &V_{1} = (I_{m},0)[A_{24}^{\dagger}(K-A_{11}V_{3}B_{22}-A_{33}W_{3}B_{44})-A_{24}^{\dagger}U_{11}B_{13}+L_{A_{24}}U_{12}],\\ &W_{1} = (0,I_{m})[A_{24}^{\dagger}(K-A_{11}V_{3}B_{22}-A_{33}W_{3}B_{44})-A_{24}^{\dagger}U_{11}B_{13}+L_{A_{24}}U_{12}],\\ &W_{2} = [R_{A_{24}}(K-A_{11}V_{3}B_{22}-A_{33}W_{3}B_{44})B_{13}^{\dagger}+A_{24}A_{24}^{\dagger}U_{11}+U_{21}R_{B_{13}}]\begin{pmatrix} 0\\I_{n} \end{pmatrix},\\ &V_{2} = [R_{A_{24}}(K-A_{11}V_{3}B_{22}-A_{33}W_{3}B_{44})B_{13}^{\dagger}+A_{24}A_{24}^{\dagger}U_{11}+U_{21}R_{B_{13}}]\begin{pmatrix} I_{n}\\0 \end{pmatrix},\quad\quad\quad\quad\\ &V_{3} = E_{11}^{\dagger}KE_{33}^{\dagger}-E_{11}^{\dagger}E_{22}N^{\dagger}KE_{33}^{\dagger}-E_{11}^{\dagger}SE_{22}^{\dagger}KM^{\dagger}E_{44}E_{33}^{\dagger}\\ &\quad\quad-E_{11}^{\dagger}SU_{31}R_{M}E_{44}E_{33}^{\dagger}+L_{E_{11}}U_{32}+U_{33}R_{E_{33}},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\\ &W_{3} = N^{\dagger}KE_{44}+S^{\dagger}SE_{22}^{\dagger}KM^{\dagger}+L_{N}L_{S}U_{41}+L_{N}U_{31}R_{M}-U_{42}R_{E_{44}}, \end{align*}$

$U_{11}, U_{12}, U_{21}, U_{31}, U_{32}, U_{33}, U_{41},$ and $U_{42}$ are arbitrary quaternion matrices with appropriate sizes, and $m$ and $n$ denote the column number of $C_{4}$ and the row number of $D_{4}$ , respectively.

3. Solvability conditions and expression of general solution for system (1.1)

Some necessary and sufficient conditions for System (1.1) to be solvable will be established in this section. The general solution of System (1.1) will also be derived in this section. Moreover, we provide an example to illustrate our main results.

Theorem 3.1. Let $A_{ii}, B_{ii}, C_{ii}, D_{ii}, E_{ii}, F_{ii}, H_{11},$ and $G_{11}$ (i = 1, 2) be given quaternion matrices. Put

$\begin{align} &\begin{aligned} \left\{\begin{array}{c} A_{1} = C_{11}L_{A_{11}},P_{1} = E_{11}-C_{11}A_{11}^{\dagger}B_{11}D_{11},B_{2} = R_{A_{22}}D_{22},\\ P_{2} = E_{22}-C_{22}B_{22}A_{22}^{\dagger}D_{22},\hat{B_{1}} = R_{B_{2}}R_{A_{22}}F_{22},\hat{A_{2}} = F_{11}L_{A_{11}}L_{A_{1}},\\ \hat{A_{3}} = {F_{11}}L_{A_{11}},\hat{B_{3}} = R_{D_{11}}H_{11},\hat{A_{4}} = L_{C_{22}},\hat{B_{4}} = R_{A_{22}}F_{22},H_{11}L_{\hat{B_{1}}} = \hat{B_{11}},\\ P = G_{11}-F_{11}A_{11}^{\dagger}B_{11}H_{11}-F_{11}L_{A_{11}}A_{1}^{\dagger}P_{1}D_{11}^{\dagger}H_{11}-B_{22}A_{22}^{\dagger}F_{22}-C_{22}^{\dagger}P_{2}B_{2}^{\dagger}R_{A_{22}}F_{22}, \end{array} \right. \end{aligned} \end{align}$

(3.1)

$\begin{align} &\begin{aligned} \left\{\begin{array}{c} \hat{B_{22}}L_{\hat{B_{11}}} = N_{1},\hat{B_{3}}L_{\hat{B_{1}}} = \hat{B_{22}},\hat{B_{4}}L_{\hat{B_{1}}} = \hat{B_{33}},R_{\hat{A_{2}}}\hat{A_{3}} = \hat{M_{1}},S_{1} = \hat{A_{3}}L_{\hat{M_{1}}}, T_1 = PL_{\hat{B_{1}}},\\ C = R_{\hat{M_{1}}}R_{\hat{A_{2}}},C_{1} = C\hat{A_{4}},C_{2} = R_{\hat{A_{2}}}\hat{A_{4}},C_{3} = R_{\hat{A_{3}}}\hat{A_{4}},C_{4} = \hat{A_{4}},D = L_{\hat{B_{11}}}L_{N_{1}},\\ D_{1} = \hat{B_{33}},D_{2} = \hat{B_{33}}L_{\hat{B_{22}}},D_{4} = \hat{B_{33}}D,E_{1} = CT_{1},E_{2} = R_{\hat{A_{2}}}T_{1}L_{\hat{B_{22}}}, \\ E_{3} = R_{\hat{A_{3}}}T_{1}L_{\hat{B_{11}}}, E_{4} = T_{1}D, \hat{C_{11}} = ( L_{C_{2}},L_{C_{4}}),D_{3} = \hat{B_{33}}L_{\hat{B_{11}}},\hat{D_{11}} = \begin{pmatrix} R_{D_{1}}\\ R_{D_{3}} \end{pmatrix},\\ \hat{C_{22}} = L_{C_{1}},\hat{D_{22}} = R_{D_{2}},\hat{C_{33}} = L_{C_{3}},\hat{D_{33}} = R_{D_{4}},\hat{E_{11}} = R_{\hat{C_{11}}}\hat{C_{22}},\\ \hat{E_{22}} = R_{\hat{C_{11}}}\hat{C_{22}},\hat{E_{33}} = \hat{D_{22}}L_{\hat{D_{11}}},\hat{E_{44}} = \hat{D_{33}}L_{\hat{D_{11}}},M = R_{\hat{E_{11}}}\hat{E_{22}},\\ N = \hat{E_{44}}L_{\hat{E_{33}}},F = F_{2}-F_{1},E = R_{\hat{C_{11}}}FL_{\hat{D_{11}}},S = \hat{E_{22}}L_{M},\hat{F_{11}} = C_{2}L_{C_{1}},\\ G_{1} = E_{2}-C_{2}C_{1}^{\dagger}E_{1}D_{1}^{\dagger}D_{2},\hat{F_{22}} = C_{4}L_{C_{3}},G_{2} = E_{4}-C_{4}C_{3}^{\dagger}E_{3}D_{3}^{\dagger}D_{4},\\ F_{1} = C_{1}^{\dagger}E_{1}D_{1}^{\dagger}+L_{C_{1}}C_{2}^{\dagger}E_{2}D_{2}^{\dagger},F_{2} = C_{3}^{\dagger}E_{3}D_{3}^{\dagger}+L_{C_{3}}C_{4}^{\dagger}E_{4}D_{4}^{\dagger}. \end{array} \right. \end{aligned} \end{align}$

(3.2)

Then, the following statements are equivalent:

$\mathrm{(1)}$ System (1.1) is solvable.

$\mathrm{(2)}$

$\begin{align*} &R_{A_{11}}B_{11} = 0,R_{A_{1}}P_{1} = 0,P_{1}L_{D_{11}} = 0,B_{22}L_{A_{22}} = 0,R_{C_{22}}P_{2} = 0,\\ &P_{2}L_{B_{2}} = 0,R_{C_{i}}E_{i} = 0,E_{i}L_{D_{i}} = 0(i = \overline{1,4}),R_{\hat{E_{11}}}EL_{\hat{E_{44}}} = 0. \end{align*}$

$\mathrm{(3)}$

$\begin{align} &\begin{aligned} &r(B_{11},A_{11}) = r(A_{11}), r\begin{pmatrix} E_{11}&C_{11}\\ B_{11}D_{11}&A_{11} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11} \end{pmatrix}, \end{aligned} \end{align}$

(3.3)

$\begin{align} &\begin{aligned} &r\begin{pmatrix} E_{11}\\D_{11} \end{pmatrix} = r(D_{11}), r\begin{pmatrix} B_{22}\\A_{22} \end{pmatrix} = r(A_{22}), \end{aligned} \end{align}$

(3.4)

$\begin{align} &\begin{aligned} & r(E_{22},C_{22}) = r(C_{22}), r\begin{pmatrix} E_{22}&C_{22}B_{22}\\D_{22}&A_{22} \end{pmatrix} = r(D_{22},A_{22}), \end{aligned} \end{align}$

(3.5)

$\begin{align} &\begin{aligned} & r\begin{pmatrix} F_{22}&0&D_{22}&A_{22}\\B_{11}H_{11}&A_{11}&0&0\\C_{22}G_{11}&C_{22}F_{11}&E_{22}&C_{22}B_{22} \end{pmatrix} = r(F_{22},D_{22},A_{22})+r\begin{pmatrix} A_{11}\\C_{22}F_{11} \end{pmatrix}, \end{aligned}\quad\quad\quad \end{align}$

(3.6)

$\begin{align} &\begin{aligned} & r\begin{pmatrix} H_{11}&0&-D_{11}&0&0\\ F_{22}&0&0&D_{22}&A_{22}\\ 0&C_{11}&E_{11}&0&0\\ 0&A_{11}&B_{11}D_{11}&0&0\\C_{22}G_{11}&C_{22}F_{11}&0&E_{22}&C_{22}B_{22} \end{pmatrix} \\ & = r\begin{pmatrix} C_{11}\\A_{11}\\C_{22}F_{11} \end{pmatrix}+r\begin{pmatrix} H_{11}&D_{11}&0&0\\F_{22}&0&D_{22}&A_{22} \end{pmatrix}, \end{aligned} \end{align}$

(3.7)

$\begin{align} &\begin{aligned} & r\begin{pmatrix} H_{11}&0&0&0\\ F_{22}&0&D_{22}&A_{22}\\0&A_{11}&0&0\\ C_{22}G_{11}&C_{22}F_{11}&E_{22}&C_{22}B_{22} \end{pmatrix} = r\begin{pmatrix} H_{11}&0&0\\F_{22}&D_{22}&A_{22} \end{pmatrix}+r\begin{pmatrix} A_{11}\\C_{22}F_{11} \end{pmatrix}, \end{aligned} \end{align}$

(3.8)

$\begin{align} &\begin{aligned} & r\begin{pmatrix} H_{11}&0&0\\F_{22}&D_{22}&A_{22}\\C_{22}G_{11}&E_{22}&C_{22}B_{22} \end{pmatrix} = r\begin{pmatrix} H_{11}&0&0\\F_{22}&D_{22}&A_{22} \end{pmatrix}, \end{aligned} \end{align}$

(3.9)

$\begin{align} &\begin{aligned} & r\begin{pmatrix} G_{11}&F_{11}&B_{22}\\F_{22}&0&A_{22}\\B_{11}H_{11}&A_{11}&0 \end{pmatrix} = r\begin{pmatrix} F_{11}\\A_{11} \end{pmatrix}+r(F_{22},A_{22}), \end{aligned} \end{align}$

(3.10)

$\begin{align} &\begin{aligned} & r\begin{pmatrix} G_{11}&F_{11}&0&B_{22}\\H_{11}&0&-D_{11}&0\\F_{22}&0&0&A_{22}\\0&C_{11}&E_{11}&0\\0&A_{11}&B_{11}D_{11}&0 \end{pmatrix} = r\begin{pmatrix} H_{11}&D_{11}&0\\F_{22}&0&A_{22} \end{pmatrix}+r\begin{pmatrix} F_{11}\\C_{11}\\A_{11} \end{pmatrix}, \end{aligned} \end{align}$

(3.11)

$\begin{align} &\begin{aligned} & r\begin{pmatrix} G_{11}&F_{11}&B_{22}\\H_{11}&0&0\\F_{22}&0&A_{22}\\0&A_{11}&0 \end{pmatrix} = r\begin{pmatrix} H_{11}&0\\F_{22}&A_{22} \end{pmatrix}+r\begin{pmatrix} F_{11}\\A_{11} \end{pmatrix}, \end{aligned} \end{align}$

(3.12)

$\begin{align} &\begin{aligned} & r\begin{pmatrix} G_{11}&B_{22}\\H_{11}&0\\F_{22}&A_{22} \end{pmatrix} = r\begin{pmatrix} H_{11}&0\\F_{22}&A_{22} \end{pmatrix},\end{aligned} \end{align}$

(3.13)

$\begin{align} &\begin{aligned} &r\begin{pmatrix} H_{11}&0&0&0&0&0&0&D_{11}&0\\F_{22}&0&0&0&0&D_{22}&A_{22}&0&0\\0&0&H_{11}&0&0&0&0&0&0\\0&0&F_{22}&D_{22}&A_{22}&0&0&0&0\\F_{22}&0&F_{22}&0&0&0&0&0&A_{22}\\ 0&C_{11}&0&0&0&0&0&-E_{11}&0\\0&A_{11}&0&0&0&0&0&-B_{11}D_{11}&0\\C_{22}G_{11}&C_{22}F_{11}&0&0&0&E_{22}&C_{22}B_{22}&0&0 \end{pmatrix}\\ & = r\left(\begin{array}{cccccccc} H_{11} & 0 & 0 & 0 & 0 & 0 & D_{11} & 0 \\ F_{22} & 0 & 0 & 0 & D_{22} & A_{22} & 0 & 0 \\ 0 & H_{11} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & F_{22} & D_{22} & A_{22} & 0 & 0 & 0 & 0 \\ F_{22} & F_{22} & 0 & 0 & 0 & 0 & 0 & A_{22} \end{array}\right)+r\left(\begin{array}{c} C_{11} \\ A_{11} \\ C_{22} F_{11} \end{array}\right). \end{aligned} \end{align}$

(3.14)

Proof. $(1)\Leftrightarrow (2)$ : The System (1.1) can be written as follows.

$\begin{align} \begin{aligned} A_{11}X_1 = B_{11},\ X_{2}A_{22} = B_{22},\quad\quad \end{aligned} \end{align}$

(3.15)

$\begin{align} \begin{aligned} C_{11}X_1D_{11} = E_{11},\ C_{22}X_2D_{22} = E_{22}, \end{aligned} \end{align}$

(3.16)

and

$\begin{align} F_{11}X_1H_{11}+X_2F_{22} = G_{11}. \end{align}$

(3.17)

Next, the solvability conditions and the expression for the general solutions of Eq (3.15) to Eq (3.17) are given by the following steps:

Step 1: According to Lemma 2.3 and Lemma 2.4, the system (3.15) is solvable if, and only if,

$\begin{align} R_{A_{11}}B_{11} = 0,\ B_{22}L_{A_{22}} = 0. \end{align}$

(3.18)

When condition (3.18) holds, the general solution of System (3.15) is

$\begin{align} X_1 = A_{11}^{\dagger}B_{11}+L_{A_{11}}U_1,\ X_2 = B_{22}A_{22}^{\dagger}+U_2R_{A_{22}}. \end{align}$

(3.19)

Step 2: Substituting (3.19) into (3.16) yields,

$\begin{align} A_{1}U_{1}D_{11} = P_{1},\ C_{22}U_2B_{2} = P_{2}, \end{align}$

(3.20)

where $A_{1}, P_1, B_{2}, P_{2}$ are defined by (3.1). By Lemma 2.5, the system (3.20) is consistent if, and only if,

$\begin{align} R_{A_{1}}P_{1} = 0,\ P_{1}L_{D_{11}} = 0,\ R_{C_{22}}P_{2} = 0,\ P_{2}L_{B_{2}} = 0. \end{align}$

(3.21)

When (3.21) holds, the general solution to System (3.20) is

$\begin{align} U_{1} = A_{1}^{\dagger}P_{1}D_{11}^{\dagger}+L_{A_{1}}W_{1}+W_{2}R_{D_{11}}, U_2 = C_{22}^{\dagger}P_{2}B_{2}^{\dagger}+L_{C_{22}}W_{3}+W_{4}R_{B_{2}}. \end{align}$

(3.22)

Comparing (3.22) and (3.19), hence,

$\begin{align} \begin{aligned} X_{1} = A_{11}^{\dagger}B_{11}+L_{A_{11}}A_{1}^{\dagger}P_{1}D_{11}^{\dagger}+L_{A_{11}}L_{A_{1}}W_{1}+L_{A_{11}}W_{2}R_{D_{11}},\\ X_{2} = B_{22}A_{22}^{\dagger}+C_{22}^{\dagger}P_{2}B_{2}^{\dagger}R_{A_ {22}}+L_{C_{22}}W_{3}R_{A_{22}}+W_{4}R_{B_{2}}R_{A_{22}}. \end{aligned} \end{align}$

(3.23)

Step 3: Substituting (3.23) into (3.17) yields

$\begin{align} W_{4}\hat{B_{1}}+\hat{A_{2}}W_{1}H_{11}+\hat{A_{3}}W_{2}\hat{B_{3}}+\hat{A_{4}}W_{3}\hat{B_{4}} = P, \end{align}$

(3.24)

where $\hat{B_{i}}, \hat{A_{j}}(i = \overline{1, 4}, j = \overline{2, 4})$ are defined by (3.1). It follows from Lemma 2.6 that Eq (3.24) is solvable if, and only if,

$\begin{align} R_{C_{i}}E_{i} = 0, E_{i}L_{D_{i}} = 0(i = \overline{1,4}),R_{\hat{E_{11}}}EL_{\hat{E_{44}}} = 0. \end{align}$

(3.25)

When (3.25) holds, the general solution to matrix equation (3.24) is

where $C_{i}, E_{i}, D_{i}(i = \overline{1, 4}), \hat{E_{11}}, \hat{E_{44}}$ are defined as (3.2), $T = T_{1}-\hat{A_{4}}W_{3}\hat{B_{4}}, V_{i}(i = \overline{1, 8})$ are arbitrary matrices with appropriate sizes over $\mathbb{H}$ ,

$\begin{align*} &\hat{V_{1}} = (I_{m},0)[\hat{C_{11}}^{\dagger}(F-\hat{C_{22}}V_{3}\hat{D_{22}}-\hat{C_{33}}\hat{V_{3}}\hat{D_{33}})-\hat{C_{11}}^{\dagger}U_{11}\hat{D_{11}}+L_{\hat{C_{11}}}U_{12}],\\ &V_{1} = (0,I_{m})[\hat{C_{11}}^{\dagger}(F-\hat{C_{22}}V_{3}\hat{D_{22}}-\hat{C_{33}}\hat{V_{3}}\hat{D_{33}})-\hat{C_{11}}^{\dagger}U_{11}\hat{D_{11}}+L_{\hat{C_{11}}}U_{12}],\\ &V_{2} = [R_{\hat{C_{11}}}(F-\hat{C_{22}}V_{3}\hat{D_{22}}-\hat{C_{33}}\hat{V_{3}}\hat{D_{33}})\hat{D_{11}}^{\dagger}+\hat{C_{11}}\hat{C_{11}}^{\dagger}U_{11}+U_{21}R_{\hat{D_{11}}}]\begin{pmatrix} 0 \\ I_{n} \end{pmatrix},\\ &\hat{V_{2}} = [R_{\hat{C_{11}}}(F-\hat{C_{22}}V_{3}\hat{D_{22}}-\hat{C_{33}}\hat{V_{3}}\hat{D_{33}})\hat{D_{11}}^{\dagger}+\hat{C_{11}}\hat{C_{11}}^{\dagger}U_{11}+U_{21}R_{\hat{D_{11}}}]\begin{pmatrix} I_{n} \\ 0 \end{pmatrix},\\ &\hat{V_{3}} = \hat{E_{11}}^{\dagger}F\hat{E_{33}}^{\dagger}-\hat{E_{11}}^{\dagger}\hat{E_{22}}M^{\dagger}F\hat{E_{33}}^{\dagger}-\hat{E_{11}}^{\dagger}S\hat{E_{22}}^{\dagger}FN^{\dagger}\hat{E_{44}}\hat{E_{33}}^{\dagger}-\hat{E_{11}}^{\dagger}SU_{31}R_{N}\hat{E_{44}}\hat{E_{33}}^{\dagger}\\ &\quad\quad+L_{\hat{E_{11}}}U_{32}+U_{33}R_{\hat{E_{33}}},\\ &V_{3} = M^{\dagger}F\hat{E_{44}}^{\dagger}+S^{\dagger}S\hat{E_{22}}^{\dagger}FN^{\dagger}+L_{M}L_{S}U_{41}+L_{M}U_{31}R_{N}-U_{42}R_{\hat{E_{44}}}, \end{align*}$

$U_{11}, U_{12}, U_{21}, U_{31}, U_{32}, U_{33}, U_{41},$ and $U_{42}$ are any quaternion matrices with appropriate sizes, and $m$ and $n$ denote the column number of $C_{22}$ and the row number of $A_{22}$ , respectively. We summarize that System (1.1) has a solution if, and only if, (3.18), (3.21), and (3.25) hold, i.e., the System (1.1) has a solution if, and only if, (2) holds.

$(2)\Leftrightarrow (3)$ : We prove the equivalence in two parts. In the first part, we want to show that (3.18) and (3.21) are equivalent to (3.3) to (3.5), respectively. In the second part, we want to show that (3.25) is equivalent to (3.6) to (3.14). It is easy to know that there exist $X_{1}^{0}, X_{2}^{0}, U_{1}^{0},$ and $U_{2}^{0}$ such that

$\begin{align*} &A_{11}X_{1}^{0} = B_{11},\ X_{2}^{0}A_{22} = B_{22},\\ &A_1U_{1}^{0}D_{11} = P_{1},\ C_{22}U_{2}^{0}B_{2} = P_{2} \end{align*}$

holds, where

$\begin{align*} &X_{1}^{0} = A_{11}^{\dagger}B_{11}, U_{1}^{0} = A_{1}^{\dagger}P_{1}D_{11}^{\dagger},\\ &X_{2}^{0} = B_{22}A_{22}^{\dagger}, U_{2}^{0} = C_{22}^{\dagger}P_{2}B_{2}^{\dagger}, \end{align*}$

$P_{1} = E_{11}-C_{11}X_{1}^{0}D_{11}, P_{2} = E_{22}-C_{22}X_{2}^{0}D_{22},$ and $P = G_{11}-F_{11}X_{1}^{0}H_{11}-F_{11}L_{A_{11}}U_{1}^{0}H_{11}-X_{2}^{0}F_{22}-U_{2}^{0}R_{A_{22}}F_{22}.$

Part 1: We want to show that (3.18) and (3.21) are equivalent to (3.3) to (3.5), respectively. It follows from Lemma 2.1 and elementary transformations that

$\begin{align*} &(3.18) \Leftrightarrow r(R_{A_{11}}B_{11}) = 0 \Leftrightarrow r(B_{11},A_{11}) = r(A_{11}) \Leftrightarrow (3.3),\\ &(3.21) \Leftrightarrow r(R_{A_{1}}P_{1}) = 0 \Leftrightarrow r(P_{1},A_{1}) = r(A_{1}) \Leftrightarrow r(E_{11}-C_{11}A_{11}^{\dagger}B_{11}D_{11},C_{11}L_{A_{11}})\\ &\quad\quad\quad\quad = r(C_{11}L_{A_{11}}) \Leftrightarrow r\begin{pmatrix} E_{11}&C_{11}\\B_{11}D_{11}&A_{11} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11} \end{pmatrix} \Leftrightarrow (3.3),\\ &(3.21) \Leftrightarrow r(P_{1}L_{D_{11}}) = 0 \Leftrightarrow r\begin{pmatrix} P_{1}\\D_{11} \end{pmatrix} = r(D_{11}) \Leftrightarrow r\begin{pmatrix} E_{11}-C_{11}A_{11}^{\dagger}B_{11}D_{11}\\D_{11} \end{pmatrix}\\ &\quad\quad\quad\quad = r(D_{11}) \Leftrightarrow r\begin{pmatrix} E_{11}\\D_{11} \end{pmatrix} = r(D_{11}) \Leftrightarrow (3.4),\\ &(3.18) \Leftrightarrow r(B_{22}L_{A_{22}}) = 0 \Leftrightarrow r\begin{pmatrix} B_{22}\\A_{22} \end{pmatrix} = r(A_{22}) \Leftrightarrow (3.4).\quad \quad \quad \quad\quad \quad\quad\quad\quad \end{align*}$

Similarly, we can show that (3.21) is equivalent to (3.5). Hence, (3.18) and (3.21) are equivalent to (3.3) and (3.5), respectively.

Part 2: In this part, we want to show that (3.25) is equivalent to (3.6) and (3.14). According to Lemma 2.6, we have that (3.25) is equivalent to the following:

$\begin{align} &\begin{aligned} r\begin{pmatrix} P&\hat{A_{2}}&\hat{A_{3}}&\hat{A_{4}}\\ \hat{B_{1}}&0&0&0 \end{pmatrix} = r(\hat{B_{1}})+r(\hat{A_{2}},\hat{A_{3}},\hat{A_{4}}),\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \end{aligned} \end{align}$

(3.26)

$\begin{align} &\begin{aligned} r\begin{pmatrix} P&\hat{A_{2}}&\hat{A_{4}}\\\hat{B_{3}}&0&0\\\hat{B_{1}}&0&0 \end{pmatrix} = r(\hat{A_{2}},\hat{A_{4}})+r\begin{pmatrix} \hat{B_{3}}\\\hat{B_{1}} \end{pmatrix}, \end{aligned} \end{align}$

(3.27)

$\begin{align} &\begin{aligned} r\begin{pmatrix} P&\hat{A_{3}}&\hat{A_{4}}\\H_{11}&0&0\\\hat{B_{1}}&0&0 \end{pmatrix} = r(\hat{A_{3}},\hat{A_{4}})+r\begin{pmatrix} H_{11}\\\hat{B_{1}} \end{pmatrix}, \end{aligned} \end{align}$

(3.28)

$\begin{align} &\begin{aligned} r\begin{pmatrix} P&\hat{A_{4}}\\H_{11}&0\\\hat{B_{3}}&0\\\hat{B_{1}}&0 \end{pmatrix} = r\begin{pmatrix} H_{11}\\\hat{B_{3}}\\\hat{B_{1}} \end{pmatrix}+r(\hat{A_{4}}), \end{aligned} \end{align}$

(3.29)

$\begin{align} &\begin{aligned} r\begin{pmatrix} P&\hat{A_{2}}&\hat{A_{3}}\\\hat{B_{4}}&0&0\\\hat{B_{1}}&0&0 \end{pmatrix} = r(\hat{A_{2}},\hat{A_{3}})+r\begin{pmatrix} \hat{B_{4}}\\\hat{B_{1}} \end{pmatrix}, \end{aligned} \end{align}$

(3.30)

$\begin{align} &\begin{aligned} r\begin{pmatrix} P&\hat{A_{2}}\\\hat{B_{3}}&0\\\hat{B_{4}}&0\\\hat{B_{1}}&0 \end{pmatrix} = r\begin{pmatrix} \hat{B_{3}}\\\hat{B_{4}}\\\hat{B_{1}} \end{pmatrix}+r(\hat{A_{2}}), \end{aligned} \end{align}$

(3.31)

$\begin{align} &\begin{aligned} r\begin{pmatrix} P&\hat{A_{3}}\\H_{11}&0\\\hat{B_{4}}&0\\\hat{B_{1}}&0 \end{pmatrix} = r\begin{pmatrix} H_{11}\\\hat{B_{4}}\\\hat{B_{1}} \end{pmatrix}+r(\hat{A_{3}}), \end{aligned} \end{align}$

(3.32)

$\begin{align} &\begin{aligned} r\begin{pmatrix} P\\H_{11}\\\hat{B_{3}}\\\hat{B_{4}}\\\hat{B_{1}} \end{pmatrix} = r\begin{pmatrix} H_{11}\\\hat{B_{3}}\\\hat{B_{4}}\\\hat{B_{1}} \end{pmatrix}, \end{aligned} \end{align}$

(3.33)

$\begin{align} &\begin{aligned} &r\begin{pmatrix} P&\hat{A_{2}}&0&0&\hat{A_{4}}\\\hat{B_{3}}&0&0&0&0\\\hat{B_{1}}&0&0&0&0\\0&0&-P&\hat{A_{3}}&\hat{A_{4}}\\0&0&H_{11}&0&0\\0&0&\hat{B_{1}}&0&0\\\hat{B_{4}}&0&\hat{B_{4}}&0&0 \end{pmatrix} \end{aligned} = r\begin{pmatrix} \hat{B_{3}}&0\\\hat{B_{1}}&0\\0&H_{11}\\0&\hat{B_{1}}\\\hat{B_{4}}&\hat{B_{4}} \end{pmatrix}+r\begin{pmatrix} \hat{A_{2}}&0&\hat{A_{4}}\\0&\hat{A_{3}}&\hat{A_{4}} \end{pmatrix}, \end{align}$

(3.34)

respectively. Hence, we only prove that (3.26)–(3.34) are equivalent to (3.6)–(3.14) when we prove that (3.25) is equivalent to (3.6)–(3.14). Now, we prove that (3.26)–(3.34) are equivalent to (3.6)–(3.14). In fact, we only prove that (3.26), (3.30), and (3.34) are equivalent to (3.6), (3.10), and (3.14); the remaining part can be proved similarly. According to Lemma 2.1 and elementary transformations, we have that

$\begin{align*} &(3.26) = r\begin{pmatrix} P&\hat{A_{2}}&\hat{A_{3}}&\hat{A_{4}}\\ \hat{B_{1}}&0&0&0 \end{pmatrix} = r(\hat{B_{1}})+r(\hat{A_{2}},\hat{A_{3}},\hat{A_{4}}) \\ &\Leftrightarrow r\begin{pmatrix} G_{11}-F_{11}X_{1}^{0}H_{11}-F_{11}L_{A_{11}}U_{1}^{0}H_{11}-X_{2}^{0}F_{22}-U_{2}^{0}R_{A_{22}}F_{22}&F_{11}L_{A_{11}}L_{A_{1}}&F_{11}L_{A_{11}}&L_{C_{22}}\\R_{B_{2}}R_{A_{22}}F_{22}&0&0&0 \end{pmatrix}\\ & = r(R_{B_{2}}R_{A_{22}}F_{22})+r(F_{11}L_{A_{11}}L_{A_{1}},F_{11}L_{A_{11}},L_{C_{22}}) \\ &\Leftrightarrow\\ &r\begin{pmatrix} G_{11}-F_{11}X_{1}^{0}H_{11}-X_{2}^{0}F_{22}-U_{2}^{0}R_{A_{22}}F_{22}&F_{11}&I&0\\R_{A_{22}}F_{22}&0&0&B_{2}\\0&A_{11}&0&0\\0&0&C_{22}&0 \end{pmatrix} = r(R_{A_{22}}F_{22},B_{2})+r\begin{pmatrix} F_{11}&I\\A_{11}&0\\0&C_{22} \end{pmatrix}\\ &\Leftrightarrow r\begin{pmatrix} G_{11}&F_{11}&I&U_{2}^{0}B_{2}&0\\ F_{22}&0&0&B_{2}&A_{22}\\B_{11}H_{11}&A_{11}&0&0&0\\ C_{22}X_{2}^{0}F_{22}&0&C_{22}&0&0 \end{pmatrix} = r(F_{22},D_{22},A_{22})+r\begin{pmatrix} F_{11}&I\\A_{11}&0\\0&C_{22} \end{pmatrix}\\ &\Leftrightarrow r\begin{pmatrix} F_{22}&0&D_{22}&A_{22}\\B_{11}H_{11}&A_{11}&0&0\\C_{22}G_{11}&C_{22}F_{11}&E_{22}&C_{22}B_{22} \end{pmatrix} = r(F_{22},D_{22},A_{22})+r\begin{pmatrix} A_{11}\\F_{11}C_{22} \end{pmatrix} \Leftrightarrow (3.6). \end{align*}$

Similarly, we have that $(3.27) \Leftrightarrow (3.7), (3.28) \Leftrightarrow (3.8), (3.29) \Leftrightarrow (3.9)$ .

$\begin{align*} &(3.30) = r\begin{pmatrix} P&\hat{A_{2}}&\hat{A_{3}}\\\hat{B_{4}}&0&0\\\hat{B_{1}}&0&0 \end{pmatrix} = r(\hat{A_{2}},\hat{A_{3}})+r\begin{pmatrix} \hat{B_{4}}\\\hat{B_{1}} \end{pmatrix}\\ &\Leftrightarrow r\begin{pmatrix} G_{11}-F_{11}X_{1}^{0}H_{11}-F_{11}L_{A_{11}}U_{1}^{0}H_{11}-X_{2}^{0}F_{22}-U_{2}^{0}R_{A_{22}}F_{22}&F_{11}L_{A_{11}}L_{A_{1}}&F_{11}L_{A_{1}}\\R_{A_{22}}F_{22}&0&0\\R_{B_{2}}R_{A_{22}}F_{22}&0&0 \end{pmatrix}\quad\quad\quad\quad\\ & = r(F_{11}L_{A_{11}}L_{A_{1}},F_{11}L_{A_{11}})+r\begin{pmatrix} R_{A_{22}}F_{22}\\R_{B_{2}}R_{A_{22}}F_{22} \end{pmatrix}\\ &\Leftrightarrow r\begin{pmatrix} G_{11}-F_{11}X_{1}^{0}H_{11}&F_{11}&B_{22}\\ F_{22}&0&A_{22}\\ 0&A_{11}&0 \end{pmatrix} = r\begin{pmatrix} F_{11}\\A_{11} \end{pmatrix}+r(F_{22},A_{22})\\ &\Leftrightarrow r\begin{pmatrix} G_{11}&F_{11}&B_{22}\\F_{22}&0&A_{22}\\B_{11}H_{11}&A_{11}&0 \end{pmatrix} = r\begin{pmatrix} F_{11}\\A_{11} \end{pmatrix}+r(F_{22},A_{22}) \Leftrightarrow (3.10).\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \end{align*}$

Similarly, we have that $(3.31) \Leftrightarrow (3.11), (3.32) \Leftrightarrow (3.12), (3.33) \Leftrightarrow (3.13).$

$\begin{align*} &(3.34) = r\begin{pmatrix} P&\hat{A_{2}}&0&0&\hat{A_{4}}\\\hat{B_{3}}&0&0&0&0\\\hat{B_{1}}&0&0&0&0\\0&0&-P&\hat{A_{3}}&\hat{A_{4}}\\0&0&H_{11}&0&0\\0&0&\hat{B_{1}}&0&0\\\hat{B_{4}}&0&\hat{B_{4}}&0&0 \end{pmatrix} = r\begin{pmatrix} \hat{B_{3}}&0\\\hat{B_{1}}&0\\0&H_{11}\\0&\hat{B_{1}}\\\hat{B_{4}}&\hat{B_{4}} \end{pmatrix}+r\begin{pmatrix} \hat{A_{2}}&0&\hat{A_{4}}\\0&\hat{A_{3}}&\hat{A_{4}} \end{pmatrix}\\ &\Leftrightarrow r\begin{pmatrix} P&F_{11}L_{A_{11}}L_{A_{1}}&0&0&L_{C_{22}}\\ R_{D_{11}}H_{11}&0&0&0&0\\ R_{B_{2}}R_{A_{22}}F_{22}&0&0&0&0\\ 0&0&-P&F_{11}L_{A_{11}}&L_{C_{22}}\\ 0&0&H_{11}&0&0\\ 0&0&R_{B_{2}}R_{A_{22}}F_{22}&0&0\\ R_{A_{22}}F_{22}&0&R_{A_{22}}F_{22}&0&0 \end{pmatrix}\\ & = r\begin{pmatrix} R_{D_{11}}H_{11}&0\\R_{B_{2}}R_{A_{22}}F_{22}&0\\ 0&H_{11}\\0&R_{B_{2}}R_{A_{22}}F_{22}\\ R_{A_{22}F_{22}}&R_{A_{22}}F_{22} \end{pmatrix}+r\begin{pmatrix} F_{11}L_{A_{11}}L_{A_{1}}&0&L_{C_{22}}\\ 0&F_{11}L_{A_{11}}&L_{C_{22}} \end{pmatrix}\quad\quad\quad\quad\quad\quad\quad\quad\quad\\ &\Leftrightarrow r\begin{pmatrix} P&F_{11}L_{A_{11}}&0&0&L_{C_{22}}&0&0&0\\ H_{11}&0&0&0&0&D_{11}&0&0\\ R_{A_{22}}F_{22}&0&0&0&0&0&B_{2}&0\\ 0&0&-G_{11}+X_{2}^{0}F_{22}+U_{2}^{0}R_{A_{22}}F_{22}&F_{11}L_{A_{11}}&L_{C_{22}}&0&0&0\\ 0&0&H_{11}&0&0&0&0&0\\ 0&0&R_{A_{22}}F_{22}&0&0&0&0&B_{2}\\ R_{A_{22}}F_{22}&0&R_{A_{22}}F_{22}&0&0&0&0&0\\ 0&A_{1}&0&0&0&0&0&0 \end{pmatrix}\\ & = r\begin{pmatrix} H_{11}&0&D_{11}&0&0\\R_{A_{22}}&0&0&B_{2}&0\\ 0&H_{11}&0&0&0\\0&R_{A_{22}}F_{22}&0&0&B_{2}\\ R_{A_{22}}F_{22}&R_{A_{22}}F_{22}&0&0&0 \end{pmatrix}+r\begin{pmatrix} F_{11}L_{A_{11}}&0&L_{C_{22}}\\ 0&F_{11}L_{A_{11}}&L_{C_{22}}\\A_{1}&0&0 \end{pmatrix}\\ &\Leftrightarrow r\begin{pmatrix} H_{11}&0&0&0&0&0&0&D_{11}&0\\F_{22}&0&0&0&0&D_{22}&A_{22}&0&0\\0&0&H_{11}&0&0&0&0&0&0\\0&0&F_{22}&D_{22}&A_{22}&0&0&0&0\\F_{22}&0&F_{22}&0&0&0&0&0&A_{22}\\ 0&C_{11}&0&0&0&0&0&-E_{11}&0\\0&A_{11}&0&0&0&0&0&-B_{11}D_{11}&0\\C_{22}G_{11}&C_{22}F_{11}&0&0&0&E_{22}&C_{22}B_{22}&0&0 \end{pmatrix}\\ & = r\begin{pmatrix} H_{11}&0&0&0&0&0&D_{11}&0\\F_{22}&0&0&0&D_{22}&A_{22}&0&0\\0&H_{11}&0&0&0&0&0&0\\0&F_{22}&D_{22}&A_{22}&0&0&0&0\\F_{22}&F_{22}&0&0&0&0&0&A_{22} \end{pmatrix}+r\begin{pmatrix} C_{11}\\A_{11}\\C_{22}F_{11} \end{pmatrix} \Leftrightarrow (3.14).\quad\quad\quad\quad\quad \end{align*}$

□

Theorem 3.2. Let System (1.1) be solvable. Then, the general solution of System (1.1) is

$\begin{align*} &X_{1} = A_{11}^{\dagger}B_{11}+L_{A_{11}}A_{1}^{\dagger}P_{1}D_{11}^{\dagger}+L_{A_{11}}L_{A_{1}}W_{1}+L_{A_{11}}W_{2}R_{D_{11}},\\ &X_{2} = B_{22}A_{22}^{\dagger}+C_{22}^{\dagger}P_{2}B_{2}^{\dagger}R_{A_{22}}+L_{C_{22}}W_{3}R_{A_{22}}+W_{4}R_{B_{2}}R_{A_{22}}, \end{align*}$

where

$\begin{align*} &W_{1} = \hat{A_{2}}^{\dagger}T\hat{B_{11}}^{\dagger}-\hat{A_{2}}^{\dagger}\hat{A_{3}}\hat{M_{1}}^{\dagger}T\hat{B_{11}}^{\dagger}-\hat{A_{2}}^{\dagger}S_{1}\hat{A_{3}}^{\dagger}TN_{1}^{\dagger}\hat{B_{22}}\hat{B_{11}}^{\dagger}\\ &\quad\quad-\hat{A_{2}}^{\dagger}S_{1}V_{4}R_{N_{1}}\hat{B_{22}}\hat{B_{11}}^{\dagger}+L_{\hat{A_{2}}}V_{5}+V_{6}R_{\hat{B_{11}}},\\ &W_{2} = \hat{M_{1}}^{\dagger}T\hat{B_{22}}^{\dagger}+S_{1}^{\dagger}S_{1}\hat{A_{3}}^{\dagger}TN_{1}^{\dagger}+L_{\hat{M_{1}}}L_{S_{1}}V_{7}+V_{8}R_{\hat{B_{22}}}+L_{\hat{M_{1}}}V_{4}R_{N_{1}},\\ &W_{3} = F_{1}+L_{C_{2}}\hat{V_{1}}+\hat{V_{2}}R_{D_{1}}+L_{C_{1}}\hat{V_{3}}R_{D_{2}},\ or\ W_{3} = F_{2}-L_{C_{4}}V_{1}-V_{2}R_{D_{3}}-L_{C_{3}}V_{3}R_{D_{4}},\\ &W_{4} = (P-\hat{A_{2}}W_{1}H_{11}-\hat{A_{3}}W_{2}\hat{B_{3}}-\hat{A_{4}}W_{3}\hat{B_{4}})\hat{B_{1}}^{\dagger}+V_{3}R_{\hat{B_{1}}},\\ &\hat{V_{1}} = (I_{m},0)[\hat{C_{11}}^{\dagger}(F-\hat{C_{22}}V_{3}\hat{D_{22}}-\hat{C_{33}}\hat{V_{3}}\hat{D_{33}})-\hat{C_{11}}^{\dagger}U_{11}\hat{D_{11}}+L_{\hat{C_{11}}}U_{12}],\\ &V_{1} = (0,I_{m})[\hat{C_{11}}^{\dagger}(F-\hat{C_{22}}V_{3}\hat{D_{22}}-\hat{C_{33}}\hat{V_{3}}\hat{D_{33}})-\hat{C_{11}}^{\dagger}U_{11}\hat{D_{11}}+L_{\hat{C_{11}}}U_{12}],\\ &V_{2} = [R_{\hat{C_{11}}}(F-\hat{C_{22}}V_{3}\hat{D_{22}}-\hat{C_{33}}\hat{V_{3}}\hat{D_{33}})\hat{D_{11}}^{\dagger}+\hat{C_{11}}\hat{C_{11}}^{\dagger}U_{11}+U_{21}R_{\hat{D_{11}}}]\begin{pmatrix} 0 \\ I_{n} \end{pmatrix},\\ &\hat{V_{2}} = [R_{\hat{C_{11}}}(F-\hat{C_{22}}V_{3}\hat{D_{22}}-\hat{C_{33}}\hat{V_{3}}\hat{D_{33}})\hat{D_{11}}^{\dagger}+\hat{C_{11}}\hat{C_{11}}^{\dagger}U_{11}+U_{21}R_{\hat{D_{11}}}]\begin{pmatrix} I_{n} \\ 0 \end{pmatrix},\\ &\hat{V_{3}} = \hat{E_{11}}^{\dagger}F\hat{E_{33}}^{\dagger}-\hat{E_{11}}^{\dagger}\hat{E_{22}}M^{\dagger}F\hat{E_{33}}^{\dagger}-\hat{E_{11}}^{\dagger}S\hat{E_{22}}^{\dagger}FN^{\dagger}\hat{E_{44}}\hat{E_{33}}^{\dagger}-\hat{E_{11}}^{\dagger}SU_{31}R_{N}\hat{E_{44}}\hat{E_{33}}^{\dagger}\\ &\quad\quad+L_{\hat{E_{11}}}U_{32}+U_{33}R_{\hat{E_{33}}},\\ &V_{3} = M^{\dagger}F\hat{E_{44}}^{\dagger}+S^{\dagger}S\hat{E_{22}}^{\dagger}FN^{\dagger}+L_{M}L_{S}U_{41}+L_{M}U_{31}R_{N}-U_{42}R_{\hat{E_{44}}}, \end{align*}$

$T = T_{1}-\hat{A_{4}}W_{3}\hat{B_{4}}, \; V_{i}(i = \overline{4, 8})$ are arbitrary matrices with appropriate sizes over $\mathbb{H}$ , $U_{11}, U_{12}, U_{21}$ , $U_{31}, U_{32}, U_{33}, U_{41}$ , and $U_{42}$ are any quaternion matrices with appropriate sizes, and $m$ and $n$ denote the column number of $C_{22}$ and the row number of $A_{22}$ , respectively.

Next, we consider a special case of the System (1.1).

Corollary 3.3. ^[16] Let $A_{ii}, B_{ii}, C_{ii}, D_{ii}, E_{ii}, F_{ii}(i = 1, 2),$ and $G_{11}$ be given matrices with appropriate dimensions over $\mathbb{H}$ . Denote

$\begin{align*} &T = C_{11}L_{A_{11}}, K = R_{A_{22}}D_{22},\ B_{1} = R_{K}R_{A_{22}}F_{22},A_{1} = F_{11}L_{A_{11}}L_{T}, C_{3} = F_{11}L_{A_{11}},D_{3} = R_{D_{11}},\\ &C_{4} = L_{C_{22}},D_{4} = R_{A_{22}}F_{22},A_{\alpha} = R_{A_1}C_3, B_{\beta} = D_3L_{B_1},C_{c} = R_{A_{\alpha}}C_4, D_{d} = D_4L_{B_1}, E = R_{A_1}E_1L_{B_1},\\ &A = A_{11}^{\dagger}B_{11}+L_{A_{11}}T^{\dagger}(E_{11}-C_{11}A_{11}^{\dagger}B_{11}D_{11})D^{\dagger},B = B_{22}A_{22}^{\dagger}+C_{22}^{\dagger}(E_{22}-C_{22}B_{22}A_{22}^{\dagger}D_{22})K^{\dagger}R_{A_{22}},\\ &E_{1} = G_{11}-F_{11}A-BF_{22},M = R_{A_{\alpha}}C_{c},N = D_{d}L_{B_{\beta}},S = C_{c}L_{M}. \end{align*}$

Then, the following statements are equivalent:

$\mathrm{(1)}$ Equation (1.8) is consistent.

$\mathrm{(2)}$

$\begin{align*} &R_{A_{11}}B_{11} = 0, B_{22}L_{A_{22}} = 0, R_{C_{22}}E_{22} = 0, E_{11}L_{D_{11}} = 0,\\ &R_{T}(E_{11}-C_{11}A_{11}^{\dagger}B_{11}D_{11}) = 0,(E_{22}-C_{22}B_{22}A_{22}^{\dagger}D_{22})L_{K} = 0,\\ &R_{M}R_{A_{\alpha}}E = 0, EL_{B_{\beta}}L_{N} = 0, R_{A_{\alpha}}EL_{D_{d}} = 0, R_{C_c}EL_{B_{\beta}} = 0. \end{align*}$

$\mathrm{(3)}$

$\begin{align*} &r(B_{11},A_{11}) = r(A_{11}), r\begin{pmatrix} E_{11}&C_{11}\\ B_{11}D_{11}&A_{11} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11} \end{pmatrix},\quad\quad\quad\quad\quad\\ &r\begin{pmatrix} E_{11}\\D_{11} \end{pmatrix} = r(D_{11}), r\begin{pmatrix} B_{22}\\A_{22} \end{pmatrix} = r(A_{22}),\\ &r(E_{22},C_{22}) = r(C_{22}), r\begin{pmatrix} E_{22}&C_{22}B_{22}\\D_{22}&A_{22} \end{pmatrix} = r(D_{22},A_{22}),\\ &r\begin{pmatrix} F_{22}&0&D_{22}&A_{22}\\B_{11}&A_{11}&0&0\\C_{22}G_{11}&C_{22}F_{11}&E_{22}&C_{22}B_{22} \end{pmatrix} = r(F_{22},D_{22},A_{22})+r\begin{pmatrix} A_{11}\\C_{22}F_{11} \end{pmatrix},\quad\quad\\ &r\begin{pmatrix} 0&F_{22}D_{11}&D_{22}&A_{22}\\C_{11}&E_{11}&0&0\\A_{11}&B_{11}D_{11}&0&0\\ C_{22}F_{11}&C_{22}G_{11}D_{11}&E_{22}&C_{22}B_{22} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11}\\C_{22}F_{11} \end{pmatrix}+r(F_{22}D_{11},D_{22},A_{22}),\\ &r\begin{pmatrix} G_{11}&F_{11}&B_{22}\\F_{22}&0&A_{22}\\B_{11}&A_{11}&0 \end{pmatrix} = r\begin{pmatrix} F_{11}\\A_{11} \end{pmatrix}+r(F_{22},A_{22}),\\ &r\begin{pmatrix} F_{11}&G_{11}D_{11}&B_{22}\\0&F_{22}D_{11}&A_{22}\\C_{11}&E_{11}&0\\A_{11}&B_{11}D_{11}&0 \end{pmatrix} = r(F_{22}D_{11},A_{22}) +r\begin{pmatrix} F_{11}\\C_{11}\\A_{11} \end{pmatrix}.\quad\quad\quad\quad\quad\quad\quad\quad\quad \end{align*}$

Finally, we provide an example to illustrate the main results of this paper.

Example 3.4. Conside the matrix equation (1.1)

$\begin{align*} &A_{11} = \begin{pmatrix} a_{111}\\a_{121} \end{pmatrix},B_{11} = \begin{pmatrix} b_{111}&b_{112}\\b_{121}&b_{122} \end{pmatrix},C_{11} = \begin{pmatrix} c_{111}\\c_{121} \end{pmatrix},D_{11} = \begin{pmatrix} d_{111}\\d_{121} \end{pmatrix},\\ &E_{11} = \begin{pmatrix} e_{111}\\e_{121} \end{pmatrix}, A_{22} = \begin{pmatrix} a_{211}&a_{212} \end{pmatrix},B_{22} = \begin{pmatrix} b_{211}&b_{212}\\b_{221}&b_{222} \end{pmatrix},\\ &C_{22} = \begin{pmatrix} c_{211}&c_{212}\\c_{221}&c_{222} \end{pmatrix}, D_{22} = \begin{pmatrix} d_{211} \end{pmatrix}, E_{22} = \begin{pmatrix} e_{211}\\e_{221} \end{pmatrix},F_{11} = \begin{pmatrix} f_{111}\\f_{121} \end{pmatrix}, \\ &H_{11} = \begin{pmatrix} h_{111}&h_{112}\\h_{121}&h_{122} \end{pmatrix}, F_{22} = \begin{pmatrix} f_{211}&f_{212} \end{pmatrix},G_{11} = \begin{pmatrix} g_{111}&g_{112}\\g_{121}&g_{122} \end{pmatrix},\quad\quad\quad\quad\quad\quad\quad\quad \end{align*}$

where

$\begin{align*} &a_{111} = 0.9787+0.5005\boldsymbol{i}+0.0596\boldsymbol{j}+0.0424\boldsymbol{k}, a_{121} = 0.7127+0.4711\boldsymbol{i}+0.6820\boldsymbol{j}+0.0714\boldsymbol{k},\\&b_{111} = 0.5216+ 0.8181\boldsymbol{i}+0.7224\boldsymbol{j}+0.6596\boldsymbol{k}, b_{112} = 0.9730+0.8003\boldsymbol{i}+0.4324\boldsymbol{j}+0.0835\boldsymbol{k},\\ &b_{121} = 0.0967+0.8175\boldsymbol{i}+ 0.1499\boldsymbol{j}+0.5186\boldsymbol{k}, b_{122} = 0.6490+0.4538\boldsymbol{i} 0.8253\boldsymbol{j}+0.1332\boldsymbol{k},\\&c_{111} = 0.1734+0.8314\boldsymbol{i}+0.0605\boldsymbol{j}+0.5269\boldsymbol{k}, c_{121} = 0.3909+0.8034\boldsymbol{i}+0.3993\boldsymbol{j}+0.4168\boldsymbol{k},\\&d_{111} = 0.6569+0.2920\boldsymbol{i}+0.0159\boldsymbol{j}+0.1671\boldsymbol{k},d_{121} = 0.6280+0.4317\boldsymbol{i}+0.9841\boldsymbol{j}+0.1062\boldsymbol{k},\\&e_{111} = 0.3724+0.4897\boldsymbol{i}+0.9516\boldsymbol{j}+0.0527\boldsymbol{k},e_{121} = 0.1981+0.3395\boldsymbol{i}+0.9203\boldsymbol{j}+0.7379\boldsymbol{k},\\&a_{211} = 0.2691+0.4228\boldsymbol{i}+0.5479\boldsymbol{j}+0.9427\boldsymbol{k},a_{212} = 0.4177+0.9831\boldsymbol{i}+0.3015\boldsymbol{j}+0.7011\boldsymbol{k},\\&b_{211} = 0.6663+0.6981\boldsymbol{i}+0.1781\boldsymbol{j}+0.9991\boldsymbol{k},b_{212} = 0.0326+0.8819\boldsymbol{i}+0.1904\boldsymbol{j}+0.4607\boldsymbol{k},\\ &b_{221} = 0.5391+0.6665\boldsymbol{i}+0.1280\boldsymbol{j}+0.1711\boldsymbol{k},b_{222} = 0.5612+0.6692\boldsymbol{i}+0.3689\boldsymbol{j}+0.9816\boldsymbol{k},\\ &c_{211} = 0.1564+0.6448\boldsymbol{i}+0.1909\boldsymbol{j}+0.4820\boldsymbol{k},c_{212} = 0.5895+0.3846\boldsymbol{i}+0.2518\boldsymbol{j}+0.6171\boldsymbol{k},\\ &c_{221} = 0.8555+0.3763\boldsymbol{i}+0.4283\boldsymbol{j}+0.1206\boldsymbol{k},c_{222} = 0.2262+0.5830\boldsymbol{i}+0.2904\boldsymbol{j}+0.2653\boldsymbol{k},\\&d_{211} = 0.8244+0.9827\boldsymbol{i}+0.7302\boldsymbol{j}+0.3439\boldsymbol{k},e_{211} = 0.5847+0.9063\boldsymbol{i}+0.8178\boldsymbol{j}+0.5944\boldsymbol{k},\\ &e_{221} = 0.1078+0.8797\boldsymbol{i}+0.2607\boldsymbol{j}+0.0225\boldsymbol{k},f_{111} = 0.4253+0.1615\boldsymbol{i}+0.4229\boldsymbol{j}+0.5985\boldsymbol{k},\\&f_{121} = 0.3127+0.1788\boldsymbol{i}+0.0942\boldsymbol{j}+0.4709\boldsymbol{k},h_{111} = 0.6959+0.6385\boldsymbol{i}+0.0688\boldsymbol{j}+0.5309\boldsymbol{k},\\ &h_{112} = 0.4076+0.7184\boldsymbol{i}+0.5313\boldsymbol{j}+0.1056\boldsymbol{k},h_{121} = 0.6999+0.0336\boldsymbol{i}+0.3196\boldsymbol{j}+0.6544\boldsymbol{k},\\&h_{122} = 0.8200+0.9686\boldsymbol{i}+0.3251\boldsymbol{j}+0.6110\boldsymbol{k},f_{211} = 0.7788+0.4235\boldsymbol{i}+0.0908\boldsymbol{j}+0.2665\boldsymbol{k},\\&f_{212} = 0.1537+0.2810\boldsymbol{i}+0.4401\boldsymbol{j}+0.5271\boldsymbol{k},g_{111} = 0.4574+0.5181\boldsymbol{i}+0.6377\boldsymbol{j}+0.2407\boldsymbol{k},\\&g_{112} = 0.2891+0.6951\boldsymbol{i}+0.2548\boldsymbol{j}+0.6678\boldsymbol{k},g_{121} = 0.8754+0.9436\boldsymbol{i}+0.9577\boldsymbol{j}+0.6761\boldsymbol{k},\\&g_{122} = 0.6718+0.0680\boldsymbol{i}+0.2240\boldsymbol{j}+0.8444\boldsymbol{k}. \end{align*}$

Computing directly yields the following:

$\begin{align*} &r\begin{pmatrix} B_{11}&A_{11} \end{pmatrix} = r\begin{pmatrix} A_{11} \end{pmatrix} = 2,r\begin{pmatrix} E_{11}&C_{11}\\B_{11}D_{11}&A_{11} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11} \end{pmatrix} = 2,\\ &r\begin{pmatrix} E_{11}\\D_{11} \end{pmatrix} = r\begin{pmatrix} D_{11} \end{pmatrix} = 1,r\begin{pmatrix} B_{22}\\A_{22} \end{pmatrix} = r\begin{pmatrix} A_{22} \end{pmatrix} = 2,\\ &r\begin{pmatrix} E_{22}&C_{22} \end{pmatrix} = r\begin{pmatrix} C_{22} \end{pmatrix} = 2,r\begin{pmatrix} E_{22}&C_{22}B_{22}\\D_{22}&A_{22} \end{pmatrix} = r\begin{pmatrix} D_{22}&A_{22} \end{pmatrix} = 3,\\ &r\begin{pmatrix} F_{22}&0&D_{22}&A_{22}\\B_{11}H_{11}&A_{11}&0&0\\C_{22}G_{11}&C_{22}F_{11}&E_{22}&C_{22}B_{22} \end{pmatrix} = r\begin{pmatrix} F_{22}&D_{22}&A_{22} \end{pmatrix}+r\begin{pmatrix} A_{11}\\C_{22}F_{11} \end{pmatrix} = 5,\\ &r\begin{pmatrix} H_{11}&0&-D_{11}&0&0\\F_{22}&0&0&D_{22}&A_{22}\\0&C_{11}&E_{11}&0&0\\0&A_{11}&B_{11}D_{11}&0&0\\C_{22}G_{11}&C_{22}F_{11}&0&E_{22}&C_{22}B_{22} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11}\\C_{22}F_{11} \end{pmatrix}+r\begin{pmatrix} H_{11}&D_{11}&0&0\\F_{22}&0&D_{22}&A_{22} \end{pmatrix} = 7,\\ &r\begin{pmatrix} H_{11}&0&0&0\\F_{22}&0&D_{22}&A_{22}\\0&A_{11}&0&0\\C_{22}G_{11}&C_{22}F_{11}&E_{22}&C_{22}B_{22} \end{pmatrix} = r\begin{pmatrix} H_{11}&0&0\\F_{22}&D_{22}&A_{22} \end{pmatrix}+r\begin{pmatrix} A_{11}\\C_{22}F_{11} \end{pmatrix} = 6,\\ &r\begin{pmatrix} H_{11}&0&0\\F_{22}&D_{22}&A_{22}\\C_{22}G_{11}&E_{22}&C_{22}B_{22} \end{pmatrix} = r\begin{pmatrix} H_{11}&0&0\\F_{22}&D_{22}&A_{22} \end{pmatrix} = 5,\\ &r\begin{pmatrix} G_{11}&F_{11}&B_{22}\\F_{22}&0&A_{22}\\B_{11}H_{11}&A_{11}&0 \end{pmatrix} = r\begin{pmatrix} F_{11}\\A_{11} \end{pmatrix}+r\begin{pmatrix} F_{22},A_{22} \end{pmatrix} = 5,\\ &r\begin{pmatrix} G_{11}&F_{11}&0&B_{22}\\H_{11}&0&-D_{11}&0\\F_{22}&0&0&A_{22}\\0&C_{11}&E_{11}&0\\0&A_{11}&B_{11}D_{11}&0 \end{pmatrix} = r\begin{pmatrix} H_{11}&D_{11}&0\\F_{22}&0&A_{22} \end{pmatrix} +r\begin{pmatrix} F_{11}\\C_{11}\\A_{11} \end{pmatrix} = 6,\\ &r\begin{pmatrix} G_{11}&F_{11}&B_{22}\\H_{11}&0&0\\F_{22}&0&A_{22}\\0&A_{11}&0 \end{pmatrix} = r\begin{pmatrix} H_{11}&0\\F_{22}&A_{22} \end{pmatrix}+r\begin{pmatrix} F_{11}\\A_{11} \end{pmatrix} = 5,\ r\begin{pmatrix} G_{11}&B_{22}\\H_{11}&0\\F_{22}&A_{22} \end{pmatrix} = r\begin{pmatrix} H_{11}&0\\F_{22}&A_{22} \end{pmatrix} = 4,\\ &r\begin{pmatrix} H_{11}&0&0&0&0&0&0&D_{11}&0\\F_{22}&0&0&0&0&D_{22}&A_{22}&0&0\\0&0&H_{11}&0&0&0&0&0&0\\0&0&F_{22}&D_{22}&A_{22}&0&0&0&0\\F_{22}&0&F_{22}&0&0&0&0&0&A_{22}\\ 0&C_{11}&0&0&0&0&0&-E_{11}&0\\0&A_{11}&0&0&0&0&0&-B_{11}D_{11}&0\\C_{22}G_{11}&C_{22}F_{11}&0&0&0&E_{22}&C_{22}B_{22}&0&0 \end{pmatrix}\\ & = r\begin{pmatrix} H_{11}&0&0&0&0&0&D_{11}&0\\F_{22}&0&0&0&D_{22}&A_{22}&0&0\\0&H_{11}&0&0&0&0&0&0\\0&F_{22}&D_{22}&A_{22}&0&0&0&0\\F_{22}&F_{22}&0&0&0&0&0&A_{22} \end{pmatrix}+r\begin{pmatrix} C_{11}\\A_{11}\\C_{22}F_{11} \end{pmatrix} = 11.\quad\quad\quad\quad\quad\quad\quad\quad\quad \end{align*}$

All rank equations in (3.3) to (3.14) hold. So, according to Theorem 3.1, the system of matrix equation (1.1) has a solution. By Theorem 3.2, the solution of System (1.1) can be expressed as

$\begin{align*} X_{1} = &\begin{pmatrix} 0.4946+0.1700\boldsymbol{i}-0.1182\boldsymbol{j}-0.3692\boldsymbol{k}&0.4051-0.0631\boldsymbol{i}-0.2403\boldsymbol{j}-0.1875\boldsymbol{k} \end{pmatrix},\\ X_{2} = &\begin{pmatrix} -0.0122+0.2540\boldsymbol{i}-0.3398\boldsymbol{j}-0.3918\boldsymbol{k}\\0.7002-0.3481\boldsymbol{i}-0.2169\boldsymbol{j}+0.0079\boldsymbol{k} \end{pmatrix}. \end{align*}$

4. The solvability conditions and the general solutions to the Systems (1.2)–(1.4)

In this section, we use the same method and technique as in Theorem 3.1 to study the three systems of Eqs (1.2)–(1.4). We only present their results and omit their proof.

Theorem 4.1. Consider the matrix equation (1.2) over $\mathbb{H}$ , where $A_{ii}, B_{ii}, C_{ii}, D_{ii}, E_{ii}, F_{ii}, G_{11}$ , and $H_{11} (i = \overline{1, 2})$ are given. Put

$\begin{align*} &A_{1} = C_{11}L_{A_{11}},P_{1} = E_{11}-C_{11}A_{11}^{\dagger}B_{11}D_{11},B_{2} = R_{A_{22}}D_{22}, P_{2} = E_{22}-C_{22}B_{22}A_{22}^{\dagger}D_{22},\\ &\hat{A_{1}} = F_{11}L_{A_{11}}L_{A_{1}},\hat{A_{2}} = F_{11}L_{A_{1}},\hat{B_{2}} = R_{D_{11}},\hat{A_{3}} = H_{11}L_{C_{22}},\hat{B_{3}} = R_{A_{22}}F_{22},\hat{B_{4}} = R_{B_{2}}R_{A_{22}}F_{22},\\ &B = G_{11}-F_{11}A_{11}^{\dagger}B_{11}-F_{11}L_{A_{11}}A_{1}^{\dagger}P_{1}D_{11}^{\dagger}-H_{11}B_{22}A_{22}^{\dagger}F_{22}-H_{11}C_{22}^{\dagger}P_{2}B_{2}^{\dagger}R_{A_{22}}F_{22},R_{\hat{A_{1}}}\hat{A_{2}} = A_{12},\\&R_{\hat{A_{1}}}\hat{A_{3}} = A_{13},R_{\hat{A_{1}}}H_{11} = A_{14},\hat{B_{3}}L_{\hat{B_{2}}} = N_{1}, R_{A_{12}}A_{13} = M_{1},S_{1} = A_{13}L_{M_{1}},R_{\hat{A_{1}}}B = T_{1},\\ &C = R_{M_{1}}R_{A_{12}},\hat{C_{1}} = CA_{14}, \hat{C_{2}} = R_{A_{12}}A_{14},\hat{C_{3}} = R_{A_{13}}A_{14},\hat{C_{4}} = A_{14},D = L_{\hat{B_{2}}}L_{N_{1}},\hat{D_{1}} = \hat{B_{4}},\\ &\hat{D_{2}} = \hat{B_{4}}L_{\hat{B_{3}}},\hat{D_{3}} = \hat{B_{4}}L_{\hat{B_{2}}},\hat{D_{4}} = \hat{B_{4}}D,\hat{E_{1}} = CT_{1},\hat{E_{2}} = R_{A_{12}}T_{1}L_{\hat{B_{3}}},\hat{E_{3}} = R_{A_{13}}T_{1}L_{\hat{B_{2}}},\hat{E_{4}} = T_{1}D,\\ &C_{24} = (L_{\hat{C_{2}}},L_{\hat{C_{4}}}),D_{13} = \begin{pmatrix} R_{\hat{D_{1}}}\\ R_{\hat{D_{3}}} \end{pmatrix}, C_{12} = L_{\hat{C_{1}}},D_{12} = R_{\hat{D_{2}}},C_{33} = L_{\hat{C_{3}}},D_{33} = R_{\hat{D_{4}}},E_{24} = R_{C_{24}}C_{12},\\ &E_{13} = R_{C_{24}}C_{33},E_{33} = D_{12}L_{D_{13}},E_{44} = D_{33}L_{D_{13}},M = R_{E_{24}}E_{13},N = E_{44}L_{E_{33}},F = F_{2}-F_{1},\\ &E = R_{C_{24}}FL_{D_{13}},S = E_{13}L_{M},\hat{F_{11}} = \hat{C_{2}}L_{\hat{C_{1}}},\hat{G_{1}} = \hat{E_{2}}-\hat{C_{2}}\hat{C_{1}}^{\dagger}\hat{E_{1}}\hat{D_{1}}^{\dagger}\hat{D_{2}},F_{33} = \hat{C_{4}}L_{\hat{C_{3}}},\\ &\hat{G_{2}} = \hat{E_{4}}-\hat{C_{4}}\hat{C_{3}}^{\dagger}\hat{E_{3}}\hat{D_{3}}^{\dagger}\hat{D_{4}},F_{1} = \hat{C_{1}}^{\dagger}\hat{E_{1}}\hat{D_{1}}^{\dagger}+L_{\hat{C_{1}}}\hat{C_{2}}^{\dagger}\hat{E_{2}}\hat{D_{2}}^{\dagger},F_{2} = \hat{C_{3}}^{\dagger}\hat{E_{3}}\hat{D_{3}}^{\dagger}+L_{\hat{C_{3}}}\hat{C_{4}}^{\dagger}\hat{E_{4}}\hat{D_{4}}^{\dagger}. \end{align*}$

Then, the following statements are equivalent:

$\mathrm{(1)}$ System (1.2) is consistent.

$\mathrm{(2)}$

$\begin{align*} &R_{A_{11}}B_{11} = 0,R_{A_{1}}P_{1} = 0,P_{1}L_{D_{11}} = 0,B_{22}L_{A_{22}} = 0,R_{C_{22}}P_{2} = 0,\\ &P_{2}L_{B_{2}} = 0,R_{\hat{C_{i}}}\hat{E_{i}} = 0,\hat{E_{i}}L_{\hat{D_{i}}} = 0(i = \overline{1,4}),R_{E_{24}}EL_{E_{44}} = 0. \end{align*}$

$\mathrm{(3)}$

$\begin{align*} &r(B_{11},A_{11}) = r(A_{11}), r\begin{pmatrix} E_{11}&C_{11}\\ B_{11}D_{11}&A_{11} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11} \end{pmatrix},r\begin{pmatrix} E_{11}\\D_{11} \end{pmatrix} = r(D_{11}), \\ &r(B_{22},A_{22}) = r(A_{22}),r(E_{22},C_{22}) = r(C_{22}), r\begin{pmatrix} E_{22}&C_{22}B_{22}\\D_{22}&A_{22} \end{pmatrix} = r(D_{22},A_{22}),\\ &r\begin{pmatrix} G_{11}D_{11}&F_{11}&H_{11}\\E_{11}&C_{11}&0\\B_{11}D_{11}&A_{11}&0 \end{pmatrix} = r\begin{pmatrix} F_{11}&H_{11}\\C_{11}&0\\A_{11}&0 \end{pmatrix},\\ &r\begin{pmatrix} G_{11}D_{11}&F_{11}&H_{11}&0\\F_{22}D_{11}&0&0&A_{22}\\E_{11}&C_{11}&0&0\\B_{11}D_{11}&A_{11}&0&0 \end{pmatrix} = r(F_{22},A_{22})+r\begin{pmatrix} F_{11}&H_{11}\\C_{11}&0\\A_{11}&0 \end{pmatrix},\\ &r\begin{pmatrix} H_{11}&F_{11}&G_{11}D_{11}\\0&C_{11}&E_{11}\\0&A_{11}&B_{11}D_{11} \end{pmatrix} = r\begin{pmatrix} H_{11}&F_{11}\\0&C_{11}\\0&A_{11} \end{pmatrix},\\ &r\begin{pmatrix} H_{11}&F_{11}&0&G_{11}D_{11}\\0&0&A_{22}&F_{22}D_{11}\\0&C_{11}&0&E_{11}\\0&A_{11}&0&B_{11}D_{11} \end{pmatrix} = r(F_{22}D_{11},A_{22})+r\begin{pmatrix} H_{11}&F_{11}\\0&C_{11}\\0&A_{11} \end{pmatrix},\\ &r\begin{pmatrix} G_{11}D_{11}&F_{11}&H_{11}&0&0\\F_{22}D_{11}&0&0&D_{22}&A_{22}\\E_{11}&C_{11}&0&0&0&\\0&0&C_{22}&-E_{22}&-C_{22}B_{22}\\B_{11}D_{11}&A_{11}&0&0&0 \end{pmatrix} = r\begin{pmatrix} F_{11}&H_{11}\\C_{11}&0\\0&C_{22}\\A_{11}&0 \end{pmatrix}+r(F_{22},D_{22},A_{22}),\quad\quad\quad\quad\\ &r\begin{pmatrix} G_{11}D_{11}&F_{11}&H_{11}B_{22}\\F_{22}D_{11}&0&A_{22}\\E_{11}&C_{11}&0\\B_{11}D_{11}&A_{11}&0 \end{pmatrix} = r\begin{pmatrix} F_{11}\\C_{11}&A_{11} \end{pmatrix}+r(F_{22},A_{22}),\\ &r\begin{pmatrix} H_{11}&F_{11}&0&0&G_{11}D_{11}\\0&0&D_{22}&A_{22}&F_{22}D_{11}\\0&C_{11}&0&0&E_{11}\\0&A_{11}&0&0&B_{11}D_{11}\\C_{22}&0&-E_{22}&-C_{22}B_{22}&0 \end{pmatrix} = r\begin{pmatrix} H_{11}&F_{11}\\0&C_{11}\\0&A_{11}\\C_{22}&0 \end{pmatrix}+r(D_{22},A_{22},F_{22}D_{11}),\\ &r\begin{pmatrix} F_{11}&H_{11}B_{22}&G_{11}D_{11}\\0&A_{22}&F_{22}D_{11}\\C_{11}&0&E_{11}\\A_{11}&0&B_{11}D_{11} \end{pmatrix} = r\begin{pmatrix} F_{11}\\C_{11}\\A_{11} \end{pmatrix}+r (A_{22},F_{22}D_{11}),\\ &r\begin{pmatrix} G_{11}&F_{11}&0&0&H_{11}&0&0&H_{5}B_{22}&0\\F_{22}&0&0&0&0&0&0&A_{22}&0\\0&0&H_{11}&F_{11}&H_{11}&0&-H_{11}B_{22}&0&G_{11}D_{11}\\0&0&0&0&0&D_{22}&A_{22}&0&-F_{22}D_{11}\\ 0&0&C_{22}&0&0&E_{22}&0&0&0\\0&0&0&C_{11}&0&0&0&0&E_{11}\\0&0&0&A_{11}&0&0&0&0&B_{11}D_{11}\\B_{11}&A_{11}&0&0&0&0&0&0&0 \end{pmatrix}\\ & = r\begin{pmatrix} F_{22}&0&0&A_{22}&0\\0&D_{22}&A_{22}&0&F_{22}D_{11} \end{pmatrix}+r\begin{pmatrix} F_{11}&0&0&H_{11}\\0&H_{11}&F_{11}&H_{11}\\0&C_{22}&0&0\\0&0&C_{11}&0\\0&0&A_{11}&0\\A_{11}&0&0&0 \end{pmatrix}.\quad\quad\quad\quad\quad\quad\quad\quad \end{align*}$

Under these conditions, the general solution of System (1.2) is

where

$\begin{align*} &W_{1} = \hat{A_{1}}^{\dagger}(B-\hat{A_{2}}W_{1}\hat{B_{2}}-\hat{A_{3}}W_{3}\hat{B_{3}}-H_{11}W_{4}\hat{B_{4}})+L_{\hat{A_{1}}}U_{1},\quad\quad\quad\quad\quad\quad\quad\\ &W_{2} = A_{12}^{\dagger}T\hat{B_{2}}^{\dagger}-A_{12}^{\dagger}A_{13}M_{1}^{\dagger}T\hat{B_{2}}^{\dagger}-A_{12}^{\dagger}S_{1}A_{13}^{\dagger}TN_{1}^{\dagger}\hat{B_{3}}\hat{B_{2}}^{\dagger}\\ &\quad\quad-A_{12}^{\dagger}S_{1}U_{2}R_{N_{1}}\hat{B_{3}}\hat{B_{2}}^{\dagger}+L_{A_{12}}U_{3}+U_{4}R_{\hat{B_{2}}},\\ &W_{3} = M_{1}^{\dagger}T\hat{B_{3}}^{\dagger}+S_{1}^{\dagger}S_{1}A_{13}^{\dagger}TN_{1}^{\dagger}+L_{M_{1}}L_{S_{1}}U_{5}+U_{6}R_{\hat{B_{3}}}+L_{M_{1}}U_{2}R_{N_{1}},\\ &W_{4} = F_{1}+L_{\hat{C_{2}}}V_{1}+V_{2}R_{\hat{D_{1}}}+L_{\hat{C_{1}}}V_{3}R_{\hat{D_{2}}},\ or\ W_{4} = F_{2}-L_{\hat{C_{4}}}\hat{V_{1}}-\hat{V_{2}}R_{\hat {D_{3}}}-L_{\hat{C_{3}}}\hat{V_{3}}R_{\hat{D_{4}}}, \end{align*}$

where $T = T_{1}-H_{11}W_{4}\hat{B_{4}}, U_{i}(i = \overline{1, 6})$ are arbitrary matrices with appropriate sizes over $\mathbb{H}$ ,

$\begin{align*} &V_{1} = (I_{m},0)[C_{24}^{\dagger}(F-C_{12}V_{3}D_{12}-C_{33}\hat{V_{3}}D_{33})-C_{24}^{\dagger}U_{11}D_{13}+L_{C_{24}}U_{12}],\\&\hat{V_{1}} = (0,I_{m})[C_{24}^{\dagger}(F-C_{12}V_{3}D_{12}-C_{33}\hat{V_{3}}D_{33})-C_{24}^{\dagger}U_{11}D_{13}+L_{C_{24}}U_{12}], \quad \quad \quad \quad \quad \quad\\ &\hat{V_{2}} = [R_{C_{24}}(F-C_{12}V_{3}D_{12}-C_{33}\hat{V_{3}}D_{33})D_{13}^{\dagger}+C_{24}C_{24}^{\dagger}U_{11}+U_{21}R_{D_{13}}]\begin{pmatrix} 0\\ I_{n} \end{pmatrix},\\ &V_{2} = [R_{C_{24}}(F-C_{12}V_{3}D_{12}-C_{33}\hat{V_{3}}D_{33})D_{13}^{\dagger}+C_{24}C_{24}^{\dagger}U_{11}+U_{21}R_{D_{13}}]\begin{pmatrix} I_{n}\\ 0 \end{pmatrix},\\ &V_{3} = E_{24}^{\dagger}FE_{33}^{\dagger}-E_{24}^{\dagger}E_{13}M^{\dagger}FE_{33}^{\dagger}-E_{24}^{\dagger}SE_{13}^{\dagger}FN^{\dagger}E_{44}E_{33}^{\dagger}\\ &\quad\quad-E_{24}^{\dagger}SU_{31}R_{N}E_{44}E_{33}^{\dagger}+L_{E_{24}}U_{32}+U_{33}R_{E_{33}},\\ &\hat{V_{3}} = M^{\dagger}FE_{44}^{\dagger}+S^{\dagger}SE_{13}^{\dagger}FN^{\dagger}+L_{M}L_{S}U_{41}+L_{M}U_{31}R_{N}-U_{42}R_{E_{44}},\quad\quad\quad\quad\quad\quad\quad \end{align*}$

$U_{11}, U_{12}, U_{21}, U_{31}, U_{32}, U_{33}, U_{41},$ and $U_{42}$ are any quaternion matrices with appropriate sizes, and $m$ and $n$ denote the column number of $H_{11}$ and the row number of $A_{22}$ , respectively.

Theorem 4.2. Consider the matrix equation (1.3) over $\mathbb{H}$ , where $A_{ii}, B_{ii}, C_{ii}, D_{ii}, E_{ii}, F_{ii}, G_{11} \; H_{11} (i = \overline{1, 2})$ are given. Put

$\begin{align*} &A_{1} = C_{11}L_{A_{11}},P_{1} = E_{11}-C_{11}A_{11}^{\dagger}B_{11}D_{11},A_{2} = C_{22}L_{A_{22}},P_{2} = E_{22}-C_{22}A_{22}^{\dagger}B_{22}D_{22},\\ &\hat{A_{1}} = F_{11}L_{A_{11}}L_{A_{1}},\hat{A_{2}} = F_{11}L_{A_{11}},\hat{B_{2}} = R_{D_{11}},\hat{A_{11}} = H_{11}L_{A_{22}}L_{A_{2}},\hat{A_{22}} = H_{11}L_{A_{22}},\hat{B_{4}} = R_{D_{22}}F_{22},\\ &B = G_{11}-F_{11}A_{11}^{\dagger}B_{11}-F_{11}L_{A_{11}}A_{1}^{\dagger}P_{1}D_{11}^{\dagger}-H_{11}A_{22}^{\dagger}B_{22}F_{22}-H_{11}L_{A_{22}}A_{2}^{\dagger}P_{2}D_{22}^{\dagger}F_{22},\\ &R_{\hat{A_{1}}}\hat{A_{2}} = A_{12},R_{\hat{A_{1}}}\hat{A_{11}} = A_{13},R_{\hat{A_{1}}}\hat{A_{22}} = A_{33},F_{22}L_{\hat{B_{2}}} = N_{1},R_{A_{12}}A_{13} = M_{1},S_{1} = A_{13}L_{M_{1}},\\ &R_{\hat{A_{1}}}B = T_{1},C = R_{M_{1}}R_{A_{12}},\hat{C_{1}} = CA_{33},\hat{C_{2}} = R_{A_{12}}A_{33},\hat{C_{11}} = R_{A_{13}}A_{33},\hat{C_{22}} = A_{33},\\ &D = L_{\hat{B_{2}}}L_{N_{1}},\hat{D_{1}} = \hat{B_{4}},\hat{D_{2}} = \hat{B_{4}}L_{F_{22}},\hat{D_{11}} = \hat{B_{4}}L_{\hat{B_{2}}},\hat{D_{22}} = \hat{B_{4}}D,\hat{E_{1}} = CT_{1},\\ &\hat{E_{2}} = R_{A_{12}}T_{1}L_{F_{22}},\hat{E_{11}} = R_{A_{13}}T_{1}L_{\hat{B_{2}}},\hat{E_{4}} = T_{1}D,C_{24} = (L_{\hat{C_{2}}},L_{\hat{C_{22}}}),D_{13} = \begin{pmatrix} R_{\hat{D_{1}}}\\ R_{\hat{D_{11}}} \end{pmatrix},\\ &C_{21} = L_{\hat{C_{1}}},D_{12} = R_{\hat{D_{2}}},C_{33} = L_{\hat{C_{11}}},D_{33} = R_{\hat{D_{22}}},E_{11} = R_{C_{24}}C_{21},E_{22} = R_{C_{24}}C_{33},\\ &E_{33} = D_{12}L_{D_{13}},E_{44} = D_{33}L_{D_{13}},M = R_{E_{11}}E_{22},N = E_{44}L_{E_{33}},\\ &F = F_{2}-F_{1},E = R_{C_{24}}FL_{D_{13}},S = E_{22}L_{M},\hat{F_{11}} = \hat{C_{2}}L_{\hat{C_{1}}},\\ &\hat{G_{1}} = \hat{E_{2}}-\hat{C_{2}}\hat{C_{1}}^{\dagger}\hat{E_{1}}\hat{D_{1}}^{\dagger}\hat{D_{2}},\hat{F_{22}} = \hat{C_{22}}L_{\hat{C_{11}}},\hat{G_{2}} = \hat{E_{4}}-\hat{C_{22}}\hat{C_{11}}^{\dagger}\hat{E_{11}}\hat{D_{11}}^{\dagger}\hat{D_{22}},\\ &F_{1} = \hat{C_{1}}^{\dagger}\hat{E_{1}}\hat{D_{1}}^{\dagger}+L_{\hat{C_{1}}}\hat{C_{2}}^{\dagger}\hat{E_{2}}\hat{D_{2}}^{\dagger},F_{2} = \hat{C_{11}}^{\dagger}\hat{E_{11}}\hat{D_{11}}^{\dagger}+L_{\hat{C_{11}}}\hat{C_{22}}^{\dagger}\hat{E_{4}}\hat{D_{22}}^{\dagger}. \end{align*}$

Then, the following statements are equivalent:

$\mathrm{(1)}$ System (1.3) is consistent.

$\mathrm{(2)}$

$\begin{align*} &R_{A_{11}}B_{11} = 0,R_{A_{1}}P_{1} = 0,P_{1}L_{D_{11}} = 0,R_{A_{22}}B_{22} = 0,R_{A_{2}}P_{2} = 0,P_{2}L_{D_{22}} = 0,\\ &R_{\hat{C_{i}}}\hat{E_{i}} = 0,R_{\hat{C_{11}}}\hat{E_{11}} = 0,R_{\hat{C_{22}}}\hat{E_{4}} = 0,\hat{E_{i}}L_{\hat{D_{i}}} = 0(i = \overline{1,2}),\\ &\hat{E_{11}}L_{\hat{D_{11}}} = 0, \hat{E_{4}}L_{\hat{D_{22}}} = 0,R_{E_{11}}EL_{E_{44}} = 0. \end{align*}$

$\mathrm{(3)}$

$\begin{align*} &r(B_{11},A_{11}) = r(A_{11}),r\begin{pmatrix} E_{11}&C_{11}\\ B_{11}D_{11}&A_{11} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11} \end{pmatrix},r\begin{pmatrix} E_{11}\\D_{11} \end{pmatrix} = r(D_{11}),\\ &r(B_{22},A_{22}) = r(A_{22}),r\begin{pmatrix} E_{22}&C_{22}\\B_{4}D_{22}&A_{22} \end{pmatrix} = r\begin{pmatrix} C_{22}\\A_{22} \end{pmatrix},r\begin{pmatrix} E_{22}\\D_{22} \end{pmatrix} = r(D_{22}),\\ &r\begin{pmatrix} G_{11}&F_{11}&H_{11}\\B_{11}&A_{11}&0\\B_{22}F_{22}&0&A_{22} \end{pmatrix} = r\begin{pmatrix} F_{11}&H_{11}\\A_{11}&0\\0&A_{22} \end{pmatrix},\\ &r\begin{pmatrix} G_{11}&F_{11}&H_{11}\\F_{22}&0&0\\B_{11}&A_{11}&0\\0&0&A_{22} \end{pmatrix} = r(F_{22})+r\begin{pmatrix} F_{11}&H_{11}\\A_{11}&0\\0&A_{22} \end{pmatrix},\\ &r\begin{pmatrix} H_{11}&F_{11}&G_{11}D_{11}\\A_{22}&0&B_{22}F_{22}D_{11}\\0&C_{11}&E_{11}\\0&A_{11}&B_{11}D_{11} \end{pmatrix} = r\begin{pmatrix} H_{11}&F_{11}\\0&C_{11}\\0&A_{11}\\A_{22}&0 \end{pmatrix},\\ &r\begin{pmatrix} H_{11}&F_{11}&G_{11}D_{11}\\0&0&F_{22}D_{11}\\0&C_{11}&E_{11}\\0&A_{11}&B_{11}D_{11}\\A_{22}&0&0 \end{pmatrix} = r\begin{pmatrix} H_{11}&F_{11}\\0&C_{11}\\0&A_{11}\\A_{22}&0 \end{pmatrix}+r(F_{22}D_{11}),\\ &r\begin{pmatrix} G_{11}&F_{11}&H_{11}&0\\F_{22}&0&0&D_{22}\\B_{11}&A_{11}&0&0\\0&0&C_{22}&-E_{22}\\0&0&A_{22}&-B_{22}D_{22} \end{pmatrix} = r\begin{pmatrix} F_{11}&H_{11}\\A_{11}&0\\0&C_{22}\\0&A_{22} \end{pmatrix}+r(F_{22},D_{22}),\\ &r\begin{pmatrix} G_{11}&F_{11}\\F_{22}&0\\B_{11}&A_{11} \end{pmatrix} = r\begin{pmatrix} F_{11}\\A_{11} \end{pmatrix}+r(F_{22}),\\ &r\begin{pmatrix} H_{11}&F_{11}&0&G_{11}D_{11}\\0&0&D_{22}&F_{22}D_{11}\\C_{22}&0&-E_{22}&0\\0&C_{11}&0&E_{11}\\A_{22}&0&0&B_{22}F_{22}D_{11}\\0&A_{11}&0&B_{11}D_{11} \end{pmatrix} = r\begin{pmatrix} H_{11}&F_{11}\\C_{22}&0\\0&C_{11}\\A_{22}&0\\0&A_{11} \end{pmatrix}+r(D_{22},F_{22}D_{11}),\\ &r\begin{pmatrix} F_{11}&G_{11}D_{11}\\0&F_{22}D_{11}\\C_{11}&E_{11}\\A_{11}&B_{11}D_{11} \end{pmatrix} = r\begin{pmatrix} F_{11}\\C_{11}\\A_{11} \end{pmatrix}+r(F_{22}D_{11}),\\ &r\begin{pmatrix} G_{11}&F_{11}&0&0&0&H_{11}&0\\ F_{22}&0&0&0&0&0&0\\ 0&0&-G_{11}D_{11}&H_{11}&F_{11}&H_{11}&0\\ 0&0&F_{22}D_{11}&0&0&0&B_{22}\\ B_{11}&A_{11}&0&0&0&0&0\\ 0&0&0&C_{22}&0&0&E_{22}\\ 0&0&-E_{11}&0&C_{11}&0&0\\ 0&0&-B_{22}F_{22}D_{11}&A_{22}&0&0&0\\ 0&0&-B_{11}D_{11}&0&A_{11}&0&0\\ 0&0&0&0&0&A_{22}&0 \end{pmatrix}\\ & = r\begin{pmatrix} F_{22}&0&0\\0&D_{22}&F_{22}D_{11} \end{pmatrix}+r\begin{pmatrix} F_{11}&0&0&H_{11}\\0&H_{11}&F_{11}&H_{11}\\0&C_{22}&0&0\\0&A_{22}&0&0\\0&0&C_{11}&0\\0&0&A_{11}&0\\A_{11}&0&0&0\\0&0&0&A_{22} \end{pmatrix}.\quad\quad\quad\quad\quad\quad\quad \end{align*}$

Under these conditions, the general solution of System (1.3) is

$\begin{align*} &X_{1} = A_{11}^{\dagger}B_{11}+L_{A_{11}}A_{1}^{\dagger}P_{1}D_{11}^{\dagger}+L_{A_{11}}L_{A_{1}}W_{1}+L_{A_{11}}W_{2}R_{D_{11}},\\ &X_{2} = A_{22}^{\dagger}B_{4}+L_{A_{22}}A_{2}^{\dagger}P_{2}D_{22}^{\dagger}+L_{A_{22}}L_{A_{2}}W_{3}+L_{A_{22}}W_{4}R_{D_{22}}, \end{align*}$

where

$\begin{align*} &W_{1} = \hat{A_{1}}^{\dagger}(B-\hat{A_{2}}W_{1}\hat{B_{2}}-\hat{A_{11}}W_{3}F_{22}-\hat{A_{22}}W_{4}\hat{B_{4}})+L_{\hat{A_{1}}}U_{1},\\ &W_{2} = A_{12}^{\dagger}T\hat{B_{2}}^{\dagger}-A_{12}^{\dagger}A_{13}M_{1}^{\dagger}T\hat{B_{2}}^{\dagger}-A_{12}^{\dagger}S_{1}A_{13}^{\dagger}TN_{1}^{\dagger}F_{22}\hat{B_{2}}^{\dagger}\\ &\quad\quad-A_{12}^{\dagger}S_{1}U_{2}R_{N_{1}}F_{22}\hat{B_{2}}^{\dagger}+L_{A_{12}}U_{3}+U_{4}R_{\hat{B_{2}}},\\ &W_{3} = M_{1}^{\dagger}TF_{22}^{\dagger}+S_{1}^{\dagger}S_{1}A_{13}^{\dagger}TN_{1}^{\dagger}+L_{M_{1}}L_{S_{1}}U_{5}+U_{6}R_{F_{22}}+L_{M_{1}}U_{2}R_{N_{1}},\\ &W_{4} = F_{1}+L_{\hat{C_{2}}}V_{1}+V_{2}R_{\hat{D_{1}}}+L_{\hat{C_{1}}}V_{3}R_{\hat{D_{2}}},\ or\ W_{4} = F_{2}-L_{\hat{C_{22}}}\hat{V_{1}}-\hat{V_{2}}R_{\hat {D_{11}}}-L_{\hat{C_{11}}}\hat{V_{3}}R_{\hat{D_{22}}}, \end{align*}$

where $T = T_{1}-\hat{A_{22}}W_{4}\hat{B_{4}}, U_{i}(i = \overline{1, 6})$ are arbitrary matrices with appropriate sizes over $\mathbb{H}$ ,

$\begin{align*} &V_{1} = (I_{m},0)[C_{24}^{\dagger}(F-C_{21}V_{3}D_{12}-C_{33}\hat{V_{3}}D_{33})-C_{24}^{\dagger}U_{11}D_{13}+L_{C_{24}}U_{12}],\quad \quad \quad \quad \quad \quad\\ &\hat{V_{1}} = (0,I_{m})[C_{24}^{\dagger}(F-C_{21}V_{3}D_{12}-C_{33}\hat{V_{3}}D_{33})-C_{24}^{\dagger}U_{11}D_{13}+L_{C_{24}}U_{12}],\\ &\hat{V_{2}} = [R_{C_{24}}(F-C_{21}V_{3}D_{12}-C_{33}\hat{V_{3}}D_{33})D_{13}^{\dagger}+C_{24}C_{24}^{\dagger}U_{11}+U_{21}R_{D_{13}}]\begin{pmatrix} 0\\ I_{n} \end{pmatrix},\quad\quad\quad\quad\quad\\ &V_{2} = [R_{C_{24}}(F-C_{21}V_{3}D_{12}-C_{33}\hat{V_{3}}D_{33})D_{13}^{\dagger}+C_{24}C_{24}^{\dagger}U_{11}+U_{21}R_{D_{13}}]\begin{pmatrix} I_{n}\\ 0 \end{pmatrix},\\ &V_{3} = E_{11}^{\dagger}FE_{33}^{\dagger}-E_{11}^{\dagger}E_{22}M^{\dagger}FE_{33}^{\dagger}-E_{11}^{\dagger}SE_{22}^{\dagger}FN^{\dagger}E_{44}E_{33}^{\dagger}\\ &\quad\quad-E_{11}^{\dagger}SU_{31}R_{N}E_{44}E_{33}^{\dagger}+L_{E_{11}}U_{32}+U_{33}R_{E_{33}},\\ &\hat{V_{3}} = M^{\dagger}FE_{44}^{\dagger}+S^{\dagger}SE_{22}^{\dagger}FN^{\dagger}+L_{M}L_{S}U_{41}+L_{M}U_{31}R_{N}-U_{42}R_{E_{44}},\quad\quad\quad\quad\quad\quad\quad\quad \end{align*}$

$U_{11}, U_{12}, U_{21}, U_{31}, U_{32}, U_{33}, U_{41},$ and $U_{42}$ are any matrices with appropriate sizes, and $m$ and $n$ denote the column number of $H_{11}$ and the row number of $D_{22}$ , respectively.

Theorem 4.3. Consider the matrix equation (1.4) over $\mathbb{H}$ , where $A_{ii}, B_{ii}, C_{ii}, D_{ii}, E_{ii}, F_{ii}(i = \overline{1, 2}),$ and $G_{11}$ are given. Put

$\begin{align*} &\hat{A_{1}} = C_{11}L_{A_{11}},P_{1} = E_{11}-C_{11}A_{11}^{\dagger}B_{11}D_{11},\hat{A_{2}} = C_{22}L_{A_{22}},P_{2} = E_{22}-C_{22}A_{22}^{\dagger}B_{22}D_{22},\\ &A_{5} = F_{11}L_{A_1}L_{\hat{A_{1}}},A_{6} = F_{11}L_{A_{11}},A_{7} = L_{A_{22}}L_{\hat{A_{2}}},A_{8} = L_{A_{22}},B_{5} = R_{D_{11}},B_{7} = R_{D_{22}}F_{22},\quad\quad\quad\quad\\ &B = G_{11}-F_{11}A_{11}^{\dagger}B_{11}-F_{11}L_{A_1}\hat{A_{1}}^{\dagger}P_{1}D_{11}^{\dagger}-A_{22}^{\dagger}B_{22}F_{22}-L_{A_{22}}\hat{A_{2}}^{\dagger}P_{2}D_{22}^{\dagger}F_{22},\\ &R_{A_{5}}A_{6} = A_{11},R_{A_{5}}A_{7} = A_{2},R_{A_{5}}A_{8} = A_{33},F_{22}L_{B_{5}} = N_{1},R_{A_{11}}A_{2} = M_{1},S_{1} = A_{2}L_{M_{1}},\\ &R_{A_{5}}B = T_{1},C = R_{M_{1}}R_{A_{11}},\hat{C_{1}} = CA_{33},\hat{C_{2}} = R_{A_{11}}A_{33},\hat{C_{11}} = R_{A_{2}}A_{33},\hat{C_{4}} = A_{33},\\ &D = L_{B_{5}}L_{N_{1}},\hat{D_{1}} = B_{7},\hat{D_{2}} = B_{7}L_{F_{22}},\hat{D_{3}} = B_{7}L_{B_{5}},\hat{D_{4}} = B_{7}D,\hat{E_{1}} = CT_{1},\hat{E_{2}} = R_{A_{11}}T_{1}L_{F_{22}},\\ &\hat{E_{11}} = R_{A_{2}}T_{1}L_{B_{5}},\hat{E_{4}} = T_{1}D,C_{1} = (L_{\hat{C_{2}}},L_{\hat{C_{4}}}),D_{13} = \begin{pmatrix} R_{\hat{D_{1}}}\\ R_{\hat{D_{3}}} \end{pmatrix},D_{1} = L_{\hat{C_{1}}},D_{2} = R_{\hat{D_{2}}},\\ &C_{33} = L_{\hat{C_{11}}},D_{33} = R_{\hat{D_{4}}},E_{11} = R_{C_{1}}D_{1},E_{2} = R_{C_{1}}C_{33},E_{33} = D_{2}L_{D_{13}},E_{44} = D_{33}L_{D_{13}},\\ &M = R_{E_{11}}E_{2},N = E_{44}L_{E_{33}},F = \hat{F_{2}}-\hat{F_{1}},E = R_{C_{1}}FL_{D_{13}},S = E_{2}L_{M},F_{11} = \hat{C_{2}}L_{\hat{C_{1}}},\quad\quad\\ &\hat{G_{1}} = \hat{E_{2}}-\hat{C_{2}}\hat{C_{1}}^{\dagger}\hat{E_{1}}\hat{D_{1}}^{\dagger}\hat{D_{2}},F_{33} = \hat{C_{4}}L_{\hat{C_{11}}},\hat{G_{2}} = \hat{E_{4}}-\hat{C_{4}}\hat{C_{11}}^{\dagger}\hat{E_{11}}\hat{D_{3}}^{\dagger}\hat{D_{4}},\\ &\hat{F_{1}} = \hat{C_{1}}^{\dagger}\hat{E_{1}}\hat{D_{1}}^{\dagger}+L_{\hat{C_{1}}}\hat{C_{2}}^{\dagger}\hat{E_{2}}\hat{D_{2}}^{\dagger},\hat{F_{2}} = \hat{C_{11}}^{\dagger}\hat{E_{11}}\hat{D_{3}}^{\dagger}+L_{\hat{C_{11}}}\hat{C_{4}}^{\dagger}\hat{E_{4}}\hat{D_{4}}^{\dagger}. \end{align*}$

Then, the following statements are equivalent:

$\mathrm{(1)}$ Equation (1.4) is consistent.

$\mathrm{(2)}$

$\begin{align*} &R_{A_{11}}B_{11} = 0,R_{\hat{A_{1}}}P_{1} = 0,P_{1}L_{D_{11}} = 0,R_{A_{22}}B_{22} = 0,\\ &R_{\hat{A_{2}}}P_{2} = 0,P_{2}L_{D_{22}} = 0,\ R_{\hat{C_{i}}}\hat{E_{i}} = 0,\hat{E_{i}}L_{\hat{D_{i}}} = 0(i = \overline{1,2}),\\ &R_{\hat{C_{11}}}\hat{E_{11}} = 0,R_{\hat{C_{4}}}\hat{E_{4}} = 0,\hat{E_{11}}L_{\hat{D_{3}}} = 0,\hat{E_{4}}L_{\hat{D_{4}}} = 0,R_{E_{11}}EL_{E_{44}} = 0. \end{align*}$

$\mathrm{(3)}$

$\begin{align*} &r(B_{11},A_{11}) = r(A_{11}), r\begin{pmatrix} E_{11}&C_{11}\\ B_{11}D_{11}&A_{11} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11} \end{pmatrix},\\ &r\begin{pmatrix} E_{11}\\D_{11} \end{pmatrix} = r(D_{11}),\ r(B_{22},A_{22}) = r(A_{22}),\\ &r\begin{pmatrix} E_{22}&C_{22}\\B_{22}D_{22}&A_{22} \end{pmatrix} = r\begin{pmatrix} C_{22}\\A_{22} \end{pmatrix}, r\begin{pmatrix} E_{22}\\D_{22} \end{pmatrix} = r(D_{22}),\\ &r\begin{pmatrix} B_{11}&A_{11}\\A_{22}G_{11}-B_{22}F_{22}&A_{22}F_{11} \end{pmatrix} = r\begin{pmatrix} A_{11}\\A_{22}F_{11} \end{pmatrix},\\ &r\begin{pmatrix} F_{22}&0\\B_{11}&A_{11}\\A_{22}G_{11}&A_{22}F_{11} \end{pmatrix} = r(F_{22})+r\begin{pmatrix} A_{11}\\A_{22}F_{11} \end{pmatrix},\\ &r\begin{pmatrix} C_{11}&E_{11}\\A_{11}&B_{11}D_{11}\\-A_{22}F_{11}&B_{22}F_{22}D_{11}-A_{22}G_{11}D_{11} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11}\\A_{22}F_{11} \end{pmatrix},\\ &r\begin{pmatrix} 0&F_{22}D_{11}\\C_{11}&E_{11}\\A_{11}&B_{11}D_{11}\\A_{22}F_{11}&A_{22}G_{11}D_{11} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11}\\A_{22}F_{11} \end{pmatrix}+r(F_{22}D_{11}),\\ &r\begin{pmatrix} F_{22}&0&D_{22}\\C_{22}G_{11}&C_{22}F_{11}&E_{22}\\B_{11}&A_{11}&0\\A_{22}G_{11}&A_{22}F_{11}&B_{22}D_{22} \end{pmatrix} = r(F_{22},D_{22})+r\begin{pmatrix} C_{22}F_{11}\\A_{22}F_{11}\\A_{11} \end{pmatrix},\\ &r\begin{pmatrix} G_{11}&F_{11}\\F_{22}&0\\B_{11}&A_{11} \end{pmatrix} = r\begin{pmatrix} F_{11}\\A_{11} \end{pmatrix}+r(F_{22}),\\ &r\begin{pmatrix} 0&D_{22}&F_{22}D_{11}\\C_{22}F_{11}&E_{22}&C_{22}G_{11}D_{11}\\C_{11}&0&E_{22}\\A_{22}F_{11}&0&A_{22}G_{11}D_{11}-B_{22}F_{22}D_{11}\\A_{11}&0&B_{11}D_{11} \end{pmatrix}\\ & = r\begin{pmatrix} C_{22}F_{11}\\C_{11}\\A_{22}F_{11}\\A_{11} \end{pmatrix}+r(D_{22},F_{22}D_{11}),\\ &r\begin{pmatrix} F_{11}&G_{11}D_{11}\\0&F_{22}D_{11}\\C_{11}&E_{11}\\A_{11}&B_{11}D_{11} \end{pmatrix} = r\begin{pmatrix} F_{11}\\C_{11}\\A_{11} \end{pmatrix}+r(F_{22}D_{11}),\\ \end{align*}$

$\begin{align*} & r\begin{pmatrix} F_{22}&0&0&0&0\\ 0&0&F_{22}D_{11}&0&B_{22}\\ B_{11}&A_{11}&0&0&0\\ C_{22}G_{11}&C_{22}F_{11}&C_{22}G_{11}D_{11}&-C_{22}F_{11}&E_{22}\\ 0&0&-E_{11}&C_{11}&0\\ A_{22}G_{11}&A_{22}F_{11}&A_{22}G_{11}D_{11}-B_{22}F_{22}D_{11}&-A_{22}F_{11}&0\\ 0&0&-B_{11}D_{11}&A_{11}&0\\ A_{22}G_{11}&A_{22}F_{11}&0&0&0 \end{pmatrix}\\ & = r\begin{pmatrix} F_{22}&0&0\\0&F_{22}D_{11}&D_{22} \end{pmatrix}+r\begin{pmatrix} -C_{22}F_{11}&C_{22}F_{11}\\-A_{22}F_{11}&A_{22}F_{11}\\0&C_{11}\\0&A_{11}\\A_{11}&0\\A_{11}&0\\-A_{22}F_{11}&0 \end{pmatrix}.\quad\quad\quad\quad\quad\quad \end{align*}$

Under these conditions, the general solution of System (1.4) is

$\begin{align*} &X_{1} = A_{11}^{\dagger}B_{11}+L_{A_1}\hat{A_{1}}^{\dagger}P_{1}\hat{B_{1}}^{\dagger}+L_{A_1}L_{\hat{A_{1}}}W_{1}+L_{A_1}W_{2}R_{\hat{B_{1}}},\\ &X_2 = A_2^{\dagger}B_{22}+L_{A_2}\hat{A_{2}}^{\dagger}P_{2}\hat{B_{2}}^{\dagger}+L_{A_2}L_{\hat{A_{2}}}W_{3}+L_{A_3}W_{4}R_{\hat{B_{2}}}, \end{align*}$

where

$\begin{align*} &W_{1} = A_{5}^{\dagger}(B-A_{6}W_{1}B_{5}-A_{7}W_{3}F_{22}-A_{8}W_{4}B_{7})+L_{A_{5}}U_{1},\quad \quad \quad \quad \quad \quad \quad\quad \quad \quad\quad \quad \quad\quad\quad\quad\\ &W_{2} = A_{1}^{\dagger}TB_{5}^{\dagger}-A_{1}^{\dagger}A_{2}M_{1}^{\dagger}TB_{5}^{\dagger}-A_{1}^{\dagger}S_{1}A_{2}^{\dagger}TN_{1}^{\dagger}F_{22}B_{5}^{\dagger}\\ &\quad\quad-A_{1}^{\dagger}S_{1}U_{2}R_{N_{1}}F_{22}B_{5}^{\dagger}+L_{A_{1}}U_{3}+U_{4}R_{B_{5}},\\ &W_{3} = M_{1}^{\dagger}TF_{22}^{\dagger}+S_{1}^{\dagger}S_{1}A_{2}^{\dagger}TN_{1}^{\dagger}+L_{M_{1}}L_{S_{1}}U_{5}+U_{6}R_{F_{22}}+L_{M_{1}}U_{2}R_{N_{1}},\\ &W_{4} = \hat{F_{1}}+L_{\hat{C_{2}}}V_{1}+V_{2}R_{\hat{D_{1}}}+L_{\hat{C_{1}}}V_{3}R_{\hat{D_{2}}},\ or\ W_{4} = \hat{F_{2}}-L_{\hat{C_{4}}}\hat{V_{1}}-\hat{V_{2}}R_{\hat{D_{3}}}-L_{\hat{C_{11}}}\hat{V_{3}}R_{\hat{D_{4}}},\quad\quad\quad \end{align*}$

where $T = T_{1}-A_{8}W_{4}B_{7}, U_{i}(i = \overline{1, 6})$ are arbitrary matrices with appropriate sizes over $\mathbb{H}$ ,

$\begin{align*} &V_{1} = (I_{m},0)[C_{1}^{\dagger}(F-D_{1}V_{3}D_{2}-C_{33}\hat{V_{3}}D_{33})-C_{1}^{\dagger}U_{11}D_{1}+L_{C_{1}}U_{12}],\quad \quad \quad \quad \quad\quad\quad\quad\quad\quad\quad\quad\\ &\hat{V_{1}} = (0,I_{m})[C_{1}^{\dagger}(F-D_{1}V_{3}D_{2}-C_{33}\hat{V_{3}}D_{33})-C_{1}^{\dagger}U_{11}D_{1}+L_{C_{1}}U_{12}],\\ &\hat{V_{2}} = [R_{C_{1}}(F-D_{1}V_{3}D_{2}-C_{33}\hat{V_{3}}D_{33})D_{1}^{\dagger}+C_{1}C_{1}^{\dagger}U_{11}+U_{21}R_{D_{1}}]\begin{pmatrix} 0\\ I_{n} \end{pmatrix},\\ &V_{2} = [R_{C_{1}}(F-C_{2}V_{3}D_{2}-C_{33}\hat{V_{3}}D_{33})D_{1}^{\dagger}+C_{1}C_{1}^{\dagger}U_{11}+U_{21}R_{D_{1}}]\begin{pmatrix} I_{n}\\ 0 \end{pmatrix},\\ &V_{3} = E_{11}^{\dagger}FE_{33}^{\dagger}-E_{11}^{\dagger}E_{2}M^{\dagger}FE_{33}^{\dagger}-E_{11}^{\dagger}SE_{2}^{\dagger}FN^{\dagger}E_{44}E_{33}^{\dagger}\\ &\quad\quad-E_{11}^{\dagger}SU_{31}R_{N}E_{44}E_{33}^{\dagger}+L_{E_{11}}U_{32}+U_{33}R_{E_{33}},\\ &\hat{V_{3}} = M^{\dagger}FE_{44}^{\dagger}+S^{\dagger}SE_{2}^{\dagger}FN^{\dagger}+L_{M}L_{S}U_{41}+L_{M}U_{31}R_{N}-U_{42}R_{E_{44}}, \end{align*}$

$U_{11}, U_{12}, U_{21}, U_{31}, U_{32}, U_{33}, U_{41},$ and $U_{42}$ are any quaternion matrices with appropriate sizes, and $m$ and $n$ denote the column number of $A_{22}$ and the row number of $D_{22}$ , respectively.

5. An application of the system (1.1)

In this section, we use the Lemma 2.2 and the Theorem 3.1 to study matrix equation (1.9) involving $\eta$ -Hermicity matrices.

Theorem 5.1. Let $A_{11}, B_{11}, C_{11}, E_{11}, F_{11}, F_{22},$ and $G_{11}(G_{11} = G_{11}^{\eta^{*}})$ be given matrices. Put

$\begin{align*} &A_{1} = C_{11} L_{A_{11}}, P_{1} = E_{11}-C_{11} A_{11}^{\dagger} B_{11} C_{11}^{\eta^{*}}, B_{2} = A_{1}^{\eta^{*}}, P_{2} = P_{1}^{\eta^{*}},\hat{B}_{1} = R_{B_{2}}\left(F_{22} L_{A_{11}}\right)^{\eta^{*}}, \\ &\hat{A}_{3} = F_{11} L_{A_{11}}, \hat{A}_{2} = \hat{A}_{3} L_{A_{1}}, \hat{A}_{4} = L_{C_{11}},\hat{B}_{3} = \left(F_{11} \hat{A}_{4}\right)^{\eta^{*}},\hat{B}_{4} = \left(F_{22} L_{A_{11}}\right)^{\eta^{*}}, F_{11}^{\eta^{*}} L_{\hat{B}_{1}} = \hat{B}_{11},\\ &P = G_{11}-F_{11} A_{11}^{\dagger} B_{11} F_{11}^{\eta^{*}}-\hat{A}_{3} A_{1}^{\dagger} P_{1}\left(F_{11} C_{11}^{\dagger}\right)^{\eta^{*}}-\left(F_{22} A_{11}^{\dagger} B_{11}\right)^{\eta^{*}}-C_{11}^{\dagger} P_{2} B_{2}^{\dagger} \hat{B}_{4},\hat{B}_{22} L_{B_{11}} = N_{1},\\ & \hat{B}_{3} L_{\hat{B}_{1}} = \hat{B}_{22}, \hat{B}_{4} L_{\hat{B}_{1}} = \hat{B}_{33}, R_{\hat{A}_{2}} \hat{A}_{3} = \hat{M}_{1}, S_{1} = \hat{A}_{3} L_{M_{1}},T_{1} = PL_{\hat{B_{1}}}, C = R_{M_{1}} R_{\hat{A}_{2}},C_{1} = C\hat{A}_{4}, \\ & C_{2} = R_{\hat{A}_{2}} \hat{A}_{4}, C_{3} = R_{\hat{A}_{3}} \hat{A}_{4}, C_{4} = \hat{A}_{4}, D = L_{\hat{B}_{11}} L_{N_{1}},D_{1} = \hat{B}_{33},D_{2} = \hat{B}_{33} L_{\hat{B}_{22}}, D_{4} = \hat{B}_{33} D, \\ & E_{1} = C T_{1}, E_{2} = R_{\hat{A}_{2}} T_{1} L_{\hat{B}_{11}}, E_{4} = T_{1} D,\hat{C}_{11} = \begin{pmatrix} L_{C_{2}}, L_{C_{4}} \end{pmatrix},D_{3} = \hat{B}_{33} L_{\hat{B}_{11}},\hat{D}_{11} = \begin{pmatrix} R_{D_{1}}\\R_{D_{3}} \end{pmatrix}, \\ &\hat{C}_{22} = L_{C_{1}}, \hat{D}_{22} = R_{D_{2}},\ \hat{C}_{33} = L_{C_{3}}, \hat{D}_{33} = R_{D_{4}}, \hat{E}_{11} = R_{\hat{C}_{11}} \hat{C}_{22}, \hat{E}_{22} = R_{\hat{C}_{11}} \hat{C}_{33}, \\ &\hat{E}_{33} = \hat{D}_{22} L_{\hat{D}_{11}}, \hat{E}_{44} = \hat{D}_{33} L_{\hat{D}_{11}}, M = R_{\hat{E}_{11}} \hat{E}_{22}, N = \hat{E}_{44} L_{\hat{E}_{33}},\ F = F_{2}-F_{1}, E = R_{\hat{C}_{11}} F L_{\hat{D}_{11}}, \\ &S = \hat{E}_{22} L_{M},\hat{F_{11}} = C_{2} L_{C_{1}}, G_{1} = E_{2}-C_{2} C_{1}^{\dagger} E_{1} D_{1}^{\dagger} D_{2}, \hat{F_{22}} = C_{4} L_{C_{3}}, G_{2} = E_{4}-C_{4} C_{3}^{\dagger} E_{3} D_{3}^{\dagger} D_{4}, \\ &F_{1} = C_{1}^{\dagger} E_{1} D_{1}^{\dagger}+L_{C_{1}}^{\dagger} C_{2}^{\dagger} E_{2} D_{2}^{\dagger}, F_{2} = C_{3}^{\dagger} E_{3} D_{3}^{\dagger}+L_{C_{3}} C_{4}^{\dagger} E_{4} D_{4}^{\dagger} . \end{align*}$

Then, the following statements are equivalent:

$\mathrm{(1)}$ System (1.9) is solvable.

$\mathrm{(2)}$

$R_{A_{11}} B_{11} = 0, R_{A_{1}} P_{1} = 0, P_{1}\left(R_{C_{11}}\right)^{\eta^{*}} = 0, R_{C_{i}} E_{i} = 0, E_{i} L_{D_{i}} = 0(i = \overline{1,4}), R_{\hat{E}_{11}} E L_{\hat{E}_{44}} = 0 .$

$\mathrm{(3)}$

$\begin{align*} &r(B_{11}, A_{11}) = r(A_{11}), r\begin{pmatrix} E_{11} & C_{11} \\ B_{11} C_{11}^{\eta^{*}} & A_{11} \end{pmatrix} = r\begin{pmatrix} C_{11}\\A_{11} \end{pmatrix} ,\ r\begin{pmatrix} E_{11}\\C_{11}^{\eta^{*}} \end{pmatrix} = r(C_{11}),\\ &r\begin{pmatrix} F_{22}^{\eta^{*}} & 0 & C_{11}^{\eta^{*}} & A_{11}^{\eta^{\eta^{*}}} \\ B_{11} F_{11}^{\eta^{*}} & A_{11} & 0 & 0 \\ C_{11} G_{11} & C_{11} F_{11} & E_{11}^{\eta^{*}} & C_{11} B_{11}^{\eta^{*}} \end{pmatrix} = r\left(F_{22}^{\eta^{*}}, C_{11}^{\eta^{*}}, A_{11}^{\eta^{*}}\right)+r\binom{A_{11}}{C_{11} F_{11}},\\ &r\begin{pmatrix} F_{11}^{\eta^{*}} & 0 & -C_{11}^{\eta^{*}} & 0 & 0 \\ F_{22}^{\eta^{*}} & 0 & 0 & C_{11}^{\eta^{*}} & A_{11}^{\eta^{*}} \\ 0 & C_{11} & E_{11} & 0 & 0 \\ 0 & A_{11} & B_{11} C_{11}^{\eta^{*}} & 0 & 0 \\ C_{11} G_{11} & C_{11} F_{11} & 0 & E_{11}^{\eta^{*}} & C_{11} B_{11}^{\eta^{*}} \end{pmatrix} = \\ &r\begin{pmatrix} C_{11} \\ A_{11} \\ 0 \end{pmatrix} +r\begin{pmatrix} F_{11}^{\eta^{*}} & C_{11}^{\eta^{*}} & 0 & 0 \\ F_{22}^{\eta^{*}} & 0 & C_{11}^{\eta^{*}} & A_{11}^{\eta^{*}} \end{pmatrix},\\ &r\begin{pmatrix} F_{11}^{\eta^{*}} & 0 & 0 & 0 \\ F_{22}^{\eta^{*}} & 0 & C_{11}^{\eta^{*}} & A_{11}^{\eta^{*}} \\ 0 & A_{11} & 0 & 0 \\ C_{11} G_{11} & C_{11} F_{11} & E_{11}^{\eta^{*}} & C_{11} B_{11}^{\eta^{*}} \end{pmatrix} = r\begin{pmatrix} F_{11}^{\eta^{*}} & 0 & 0 \\ F_{22}^{\eta^{*}} & C_{11}^{\eta^{*}} & A_{11}^{\eta^{*}} \end{pmatrix} +r\binom{A_{11}}{C_{11} F_{11}},\\ &r\begin{pmatrix} F_{11}^{\eta^{*}} & 0 & 0 \\ F_{22}^{\eta^{*}} & C_{11}^{\eta^{*}} & A_{11}^{\eta^{*}} \\ C_{11} G_{11} & E_{11}^{\eta^{*}} & C_{11} B_{11}^{\eta^{*}} \end{pmatrix} = r\begin{pmatrix} F_{11}^{\eta^{*}} & 0 & 0 \\ F_{22}^{\eta^{*}} & C_{11}^{\eta^{*}} & A_{11}^{\eta^{*}}, \end{pmatrix},\\ &r\begin{pmatrix} G_{11} & F_{11} & B_{11}^{\eta^{*}} \\ F_{22}^{\eta^{*}} & 0 & A_{11}^{\eta^{*}} \\ B_{11} F_{11}^{\eta^{*}} & A_{11} & 0 \end{pmatrix} = r\binom{F_{11}}{A_{11}}+r\left(F_{22}^{\eta^{*}}, A_{11}^{\eta^{*}}\right), \\ &r\begin{pmatrix} G_{11} & F_{11} & 0 & B_{11}^{\eta^{*}} \\ F_{11}^{\eta^{*}} & 0 & -C_{11}^{\eta^{*}} & 0 \\ F_{22}^{\eta^{*}} & 0 & 0 & A_{11}^{\eta^{*}} \\ 0 & C_{11} & E_{11} & 0 \\ 0 & A_{11} & B_{11} C_{11}^{\eta^{*}} & 0 \end{pmatrix} = r\begin{pmatrix} F_{11}^{\eta^{*}} & C_{11}^{\eta^{*}} & 0 \\ F_{22}^{\eta^{*}} & 0 & A_{11}^{\eta^{*}} \end{pmatrix} +r\begin{pmatrix} F_{11} \\ C_{11} \\ A_{11} \end{pmatrix},\\ &r\begin{pmatrix} G_{11} & F_{11} & B_{11}^{\eta^{*}} \\ F_{11}^{\eta^{*}} & 0 & 0 \\ F_{22}^{\eta^{*}} & 0 & A_{11}^{\eta^{*}} \\ 0 & A_{11} & 0 \end{pmatrix} = r\begin{pmatrix} F_{11}^{\eta^{*}} & 0 \\ F_{22}^{\eta^{*}} & A_{11}^{\eta^{*}} \end{pmatrix} +r\binom{F_{11}}{A_{11}}, \\ &r\begin{pmatrix} G_{11} & B_{11}^{\eta^{*}} \\ F_{11}^{\eta^{*}} & 0 \\ F_{22}^{\eta^{*}} & A_{11}^{\eta^{*}} \end{pmatrix} = r\begin{pmatrix} F_{11}^{\eta^{*}} & 0 \\ F_{22}^{\eta^{*}} & A_{11}^{\eta^{*}} \end{pmatrix},\\ &r\begin{pmatrix} F_{11}^{\eta^{*}} & 0 & 0 & 0 & 0 & 0 & 0 & C_{11}^{\eta^{*}} & 0 \\ F_{22}^{\eta^{*}} & 0 & 0 & 0 & 0 & C_{11}^{\eta^{*}} & A_{11}^{\eta^{*}} & 0 & 0 \\ 0 & 0 & F_{11}^{\eta^{*}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & F_{22}^{\eta^{*}} & C_{11}^{\eta^{*}} & A_{11}^{\eta^{*}} & 0 & 0 & 0 & 0 \\ F_{22}^{\eta^{*}} & 0 & F_{22}^{\eta^{*}} & 0 & 0 & 0 & 0 & 0 & A_{11}^{\eta^{*}} \\ 0 & C_{11} & 0 & 0 & 0 & 0 & 0 & -E_{11} & 0 \\ 0 & A_{11} & 0 & 0 & 0 & 0 & 0 & -B_{11} C_{11}^{\eta^{*}} & 0 \\ C_{11} G_{11} & C_{11} F_{11} & 0 & 0 & 0 & E_{11}^{\eta^{*}} & C_{11} B_{11}^{\eta^{*}} & 0 & 0 \end{pmatrix} \\ & = r\begin{pmatrix} F_{11}^{\eta^{*}} & 0 & 0 & 0 & 0 & 0 & C_{11}^{\eta^{*}} & 0 \\ F_{22}^{\eta^{*}} & 0 & 0 & 0 & C_{11}^{\eta^{*}} & A_{11}^{\eta^{*}} & 0 & 0 \\ 0 & F_{11}^{\eta^{*}} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & F_{22}^{\eta^{*}} & C_{11}^{\eta^{*}} & A_{11}^{\eta^{*}} & 0 & 0 & 0 & 0 \\ F_{22}^{\eta^{*}} & F_{22}^{\eta^{*}} & 0 & 0 & 0 & 0 & 0 & A_{11}^{\eta^{*}} \end{pmatrix}+r\begin{pmatrix} C_{11} \\ A_{11} \\ C_{11} F_{11} \end{pmatrix}. \end{align*}$

Proof. Evidently, the system of Eq (1.9) has a solution if and only if the following matrix equation has a solution:

$\begin{equation} \begin{aligned} &A_{11}\hat{X_1} = B_{11}, C_{11}\hat{X_1}C_{11}^{\eta^{*}} = E_{11},\\ &\hat{X_2}A_{11}^{\eta^{*}} = B_{11}^{\eta^{*}}, C_{11}\hat{X_2}C_{11}^{\eta^{*}} = E_{11}^{\eta^{*}},\\ &F_{11}X_1F_{11}^{\eta^{*}}+\hat{X_2}^{\eta^{*}}F_{22}^{\eta^{*}} = G_{11}. \end{aligned} \end{equation}$

(5.1)

If (1.9) has a solution, say, $X_1$ , then $(\hat{X_1}, \ \hat{X_2}) : = (X_1, \ X_{1}^{\eta^{*}})$ is a solution of (5.1). Conversely, if (5.1) has a solution, say $(\hat{X_1}, \ \hat{X_2})$ , then it is easy to show that (1.5) has a solution

$\begin{align*} X_1 : = \dfrac{\hat{X_1}+X_{2}^{\eta^{*}}}{2}. \end{align*}$

According to Theorem 3.1, we can deduce that this theorem holds. □

6. Conclusions

We have established the solvability conditions and the expression of the general solutions to some constrained systems (1.1)–(1.4). As an application, we have investigated some necessary and sufficient conditions for Eq (1.9) to be consistent. It should be noted that the results of this paper are valid for the real number field and the complex number field as special number fields.

Author contributions

Long-Sheng Liu, Shuo Zhang and Hai-Xia Chang: Conceptualization, formal analysis, investigation, methodology, software, validation, writing an original draft, writing a review, and editing. All authors of this article have contributed equally. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

This work is supported by the National Natural Science Foundation(No. 11601328) and Key scientific research projects of univesities in Anhui province(No. 2023AH050476).

Conflict of interest

The authors declare that they have no conflicts of interest.

References

[1]	Adam AM (2020) Susceptibility of stock market returns to international economic policy: Evidence from effective transfer entropy of Africa with the implication for open innovation. J Open Innov Technol Mark Complex 6: 71. https://doi.org/10.3390/joitmc6030071 doi: 10.3390/joitmc6030071
[2]	Agyei SK, Adam AM, Bossman A, et al. (2022a) Does volatility in cryptocurrencies drive the interconnectedness between the cryptocurrencies market? Insights from wavelets. Cogent Econ Financ 10: 1. https://doi.org/10.1080/23322039.2022.2061682 doi: 10.1080/23322039.2022.2061682
[3]	Agyei SK, Isshaq Z, Frimpong S, et al. (2021) COVID‐19 and food prices in sub‐Saharan Africa. Afr Dev Rev, 1–12. https://doi.org/10.1111/1467-8268.12525 doi: 10.1111/1467-8268.12525
[4]	Agyei SK, Owusu Junior P, Bossman A, et al. (2022b) Situated information flow between food commodity and regional equity markets: an EEMD-based transfer entropy analysis. Discrete Dyn Nat Soc 2022. https://doi.org/10.1155/2022/3938331 doi: 10.1155/2022/3938331
[5]	Antonakakis N, Chatziantoniou I, Filis G (2017) Oil shocks and stock markets: Dynamic connectedness under the prism of recent geopolitical and economic unrest. Int Rev Financ Anal 50: 1–26. https://doi.org/10.1016/j.irfa.2017.01.004 doi: 10.1016/j.irfa.2017.01.004
[6]	Antonakakis N, Gupta R, Kollias C, et al. (2017) Geopolitical risks and the oil-stock nexus over 1899–2016. Financ Res Lett 23: 165–173. https://doi.org/10.1016/j.frl.2017.07.017 doi: 10.1016/j.frl.2017.07.017
[7]	Arouri MEH, Jouini J, Nguyen DK (2012) On the impacts of oil price fluctuations on European equity markets: Volatility spillover and hedging effectiveness. Energ Econ 34: 611–617. https://doi.org/10.1016/j.eneco.2011.08.009 doi: 10.1016/j.eneco.2011.08.009
[8]	Arouri MEH, Rault C (2012) Oil prices and stock markets in gcc countries: Empirical evidence from panel analysis. Int J Financ Econ 17: 242–253. https://doi.org/10.1002/ijfe.443 doi: 10.1002/ijfe.443
[9]	Asafo-Adjei E, Adam AM, Darkwa P (2021a) Can crude oil price returns drive stock returns of oil producing countries in Africa? Evidence from bivariate and multiple wavelet. Macroecon Financ Emerg Mark Econ, 1–19. https://doi.org/10.1080/17520843.2021.1953864 doi: 10.1080/17520843.2021.1953864
[10]	Asafo-Adjei E, Agyapong D, Agyei SK, et al. (2020) Economic policy uncertainty and stock returns of Africa: A wavelet coherence analysis. Discrete Dyn Nat Soc 2020: 1–8. https://doi.org/10.1155/2020/8846507 doi: 10.1155/2020/8846507
[11]	Asafo-Adjei E, Owusu Junior P, Adam AM (2021b) Information flow between global equities and cryptocurrencies: A VMD-based entropy evaluating shocks from COVID-19 pandemic. Complexity 2021. https://doi.org/10.1155/2021/4753753 doi: 10.1155/2021/4753753
[12]	Badeeb RA, Lean HH (2018) Asymmetric impact of oil price on Islamic sectoral stocks. Energ Econ 71: 128–139. https://doi.org/10.1016/j.eneco.2017.11.012 doi: 10.1016/j.eneco.2017.11.012
[13]	Bahloul S, Khemakhem I (2021) Dynamic return and volatility connectedness between commodities and Islamic stock market indices. Resour Policy 71: 101993. https://doi.org/10.1016/j.resourpol.2021.101993 doi: 10.1016/j.resourpol.2021.101993
[14]	Bahloul S, Mroua M, Naifar N (2021) Are Islamic indexes, Bitcoin and gold, still "safe-haven" assets during the COVID-19 pandemic crisis? Int J Islamic Middle Eastern Financ Manage. https://doi.org/10.1108/IMEFM-06-2020-0295 doi: 10.1108/IMEFM-06-2020-0295
[15]	Balcilar M, Cerci G, Demirer R (2016) Is there a role for Islamic bonds in global diversification strategies? Managerial Financ 42: 656–679. https://doi.org/10.1108/MF-07-2015-0189 doi: 10.1108/MF-07-2015-0189
[16]	Baur DG, Lucey BM (2010) Is gold a hedge or a safe haven? An analysis of stocks, bonds and gold. Financ Rev 45: 217–229. https://doi.org/10.1111/j.1540-6288.2010.00244.x doi: 10.1111/j.1540-6288.2010.00244.x
[17]	Baur DG, McDermott TK (2010) Is gold a safe haven? International evidence. J Bank Financ 34: 1886–1898. https://doi.org/10.1016/j.jbankfin.2009.12.008 doi: 10.1016/j.jbankfin.2009.12.008
[18]	Baur DG, McDermott TKJ (2016) Why is gold a safe haven? J Behav Expe Financ 10: 63–71. https://doi.org/10.1016/j.jbef.2016.03.002 doi: 10.1016/j.jbef.2016.03.002
[19]	BBC (2020) Coronavirus: Oil price collapses to lowest level for 18 years. Retrieved July 5, 2021, Available from: https://www.bbc.com/news/business-52089127.
[20]	Binder M, Coad A (2011) From average Joe's happiness to miserable Jane and cheerful John: Using quantile regressions to analyze the full subjective well-being distribution. J Econ Behav Organ 79: 275–290. https://doi.org/10.1016/j.jebo.2011.02.005 doi: 10.1016/j.jebo.2011.02.005
[21]	Boateng E, Adam AM, Owusu Junior P (2021) Modelling the heterogeneous relationship between the crude oil implied volatility index and African stocks in the coronavirus pandemic. Resour Policy 74: 102389. https://doi.org/10.1016/j.resourpol.2021.102389 doi: 10.1016/j.resourpol.2021.102389
[22]	Bossman A (2021) Information flow from COVID-19 pandemic to Islamic and conventional equities: An ICEEMDAN-induced transfer entropy analysis. Complexity 2021: 1–20. https://doi.org/10.1155/2021/4917051 doi: 10.1155/2021/4917051
[23]	Bossman A, Agyei SK, Owusu Junior P, et al. (2022a) Flights-to-and-from-quality with Islamic and conventional bonds in the COVID-19 pandemic era: ICEEMDAN-based transfer entropy. Complexity 2022: 1–25. https://doi.org/10.1155/2022/1027495 doi: 10.1155/2022/1027495
[24]	Bossman A, Owusu Junior P, Tiwari AK (2022b) Dynamic connectedness and spillovers between Islamic and conventional stock markets: time- and frequency-domain approach in COVID-19 era. Heliyon 8: e09215. https://doi.org/10.1016/j.heliyon.2022.e09215 doi: 10.1016/j.heliyon.2022.e09215
[25]	Bouoiyour J, Selmi R, Wohar ME (2019) Safe havens in the face of Presidential election uncertainty: A comparison between Bitcoin, oil and precious metals. Appl Econ 1–13. https://doi.org/10.1080/00036846.2019.1645289 doi: 10.1080/00036846.2019.1645289
[26]	Bouri E, Molnár P, Azzi G, et al. (2017) On the hedge and safe haven properties of Bitcoin: Is it really more than a diversifier? Financ Res Lett 20: 192–198. https://doi.org/10.1016/j.frl.2016.09.025 doi: 10.1016/j.frl.2016.09.025
[27]	Bouri E, Shahzad SJH, Roubaud D, et al. (2020) Bitcoin, gold, and commodities as safe havens for stocks: New insight through wavelet analysis. Q Rev Econ Financ 77: 156–164. https://doi.org/10.1016/j.qref.2020.03.004 doi: 10.1016/j.qref.2020.03.004
[28]	Chang BH, Sharif A, Aman A, et al. (2020) The asymmetric effects of oil price on sectoral Islamic stocks: New evidence from quantile-on-quantile regression approach. Resour Policy 65: 101571. https://doi.org/10.1016/j.resourpol.2019.101571 doi: 10.1016/j.resourpol.2019.101571
[29]	Diaz EM, Perez de Gracia F (2017) Oil price shocks and stock returns of oil and gas corporations. Financ Res Lett 20: 75–80. https://doi.org/10.1016/j.frl.2016.09.010 doi: 10.1016/j.frl.2016.09.010
[30]	Dickey DA, Fuller WA (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49: 1057. https://doi.org/10.2307/1912517 doi: 10.2307/1912517
[31]	Enwereuzoh PA, Odei-Mensah J, Owusu Junior P (2021) Crude oil shocks and African stock markets. Res Int Bus Financ 55: 101346. https://doi.org/10.1016/j.ribaf.2020.101346 doi: 10.1016/j.ribaf.2020.101346
[32]	Fama EF (1970) Efficient Capital Markets: A Review of the Theory. J Financ 25: 383–417.
[33]	Fama EF (2021) Efficient capital markets: A review of theory and empirical work, In: J. H. Cochrane & T. J. Moskowitz (Eds.), The Fama Portfolio, Chicago: University of Chicago Press, 76–121. Available from: https://doi.org/10.7208/9780226426983-007.
[34]	Fatum R, Yamamoto Y (2016) Intra-safe haven currency behavior during the global financial crisis. J Int Money Financ 66: 49–64. https://doi.org/10.1016/j.jimonfin.2015.12.007 doi: 10.1016/j.jimonfin.2015.12.007
[35]	Gao X, Ren Y, Umar M (2021) To what extent does COVID-19 drive stock market volatility? A comparison between the U.S. and China. Econ Res-Ekonomska Istraživanja, 1–21. https://doi.org/10.1080/1331677x.2021.1906730 doi: 10.1080/1331677x.2021.1906730
[36]	Goodell JW (2020) COVID-19 and finance: Agendas for future research. Financ Res Lett 35: 101512. https://doi.org/10.1016/j.frl.2020.101512 doi: 10.1016/j.frl.2020.101512
[37]	Gorton G, Lewellen S, Metrick A (2012) The safe-asset share. Am Econ Rev 102: 101–106. https://doi.org/10.1257/aer.102.3.101 doi: 10.1257/aer.102.3.101
[38]	Gourène GAZ, Mendy P (2018) Oil prices and African stock markets co-movement: A time and frequency analysis. J Afr Trade 5: 55. https://doi.org/10.1016/j.joat.2018.03.002 doi: 10.1016/j.joat.2018.03.002
[39]	Grisse C, Nitschka T (2013) On financial risk and the safe haven characteristics of Swiss franc exchange rates. J Empir Financ 32: 153–164. https://doi.org/10.1016/j.jempfin.2015.03.006 doi: 10.1016/j.jempfin.2015.03.006
[40]	Hashmi SM, Chang BH, Rong L (2021) Asymmetric effect of COVID-19 pandemic on E7 stock indices: Evidence from quantile-on-quantile regression approach. Res Int Bus Financ 58: 101485. https://doi.org/10.1016/j.ribaf.2021.101485 doi: 10.1016/j.ribaf.2021.101485
[41]	Ijasan K, Owusu Junior P, Tweneboah G, et al. (2021) Analysing the relationship between global REITs and exchange rates: Fresh evidence from frequency-based quantile regressions. Adv Decis Sci 25: 58–91. https://doi.org/10.47654/v25y2021i3p58-91 doi: 10.47654/v25y2021i3p58-91
[42]	Ijasan K, Tweneboah G, Omane-Adjepong M, et al (2019) On the global integration of REITs market returns: A multiresolution analysis. Cogent Econ Financ 7: 1690211. https://doi.org/10.1080/23322039.2019.1690211 doi: 10.1080/23322039.2019.1690211
[43]	Jena SK, Tiwari AK, Hammoudeh S, et al. (2019) Distributional predictability between commodity spot and futures: Evidence from nonparametric causality-in-quantiles tests. Energ Econ 78: 615–628. https://doi.org/10.1016/j.eneco.2018.11.013 doi: 10.1016/j.eneco.2018.11.013
[44]	Ji Q, Zhang D, Zhao Y (2020) Searching for safe-haven assets during the COVID-19 pandemic. Int Rev Financ Anal 71: 101526. https://doi.org/10.1016/j.irfa.2020.101526 doi: 10.1016/j.irfa.2020.101526
[45]	Jouini J (2013) Stock markets in GCC countries and global factors: A further investigation. Econ Model 31: 80–86. https://doi.org/10.1016/j.econmod.2012.11.039 doi: 10.1016/j.econmod.2012.11.039
[46]	Kang W, Perez de Gracia F, Ratti RA (2017) Oil price shocks, policy uncertainty, and stock returns of oil and gas corporations. J Int Money Financ 70: 344–359. https://doi.org/10.1016/j.jimonfin.2016.10.003 doi: 10.1016/j.jimonfin.2016.10.003
[47]	Kassouri Y, Altıntaş H (2020) Threshold cointegration, nonlinearity, and frequency domain causality relationship between stock price and Turkish Lira. Res Int Bus Financ 52: 101097. https://doi.org/10.1016/J.RIBAF.2019.101097 doi: 10.1016/J.RIBAF.2019.101097
[48]	Kassouri Y, Altıntaş H (2021) The quantile dependence of the stock returns of clean" and "dirty" firms on oil demand and supply shocks. J Commodity Mark, 100238. https://doi.org/10.1016/J.JCOMM.2021.100238 doi: 10.1016/J.JCOMM.2021.100238
[49]	Kassouri Y, Altıntaş H, Bilgili F (2020) An investigation of the financial resource curse hypothesis in oil-exporting countries: The threshold effect of democratic accountability. J Multinational Financ Manage 56: 100639. https://doi.org/10.1016/J.MULFIN.2020.100639 doi: 10.1016/J.MULFIN.2020.100639
[50]	Kelikume I, Muritala O (2019) The impact of changes in oil price on stock market: Evidence from Africa. Int J Manage. Econ Soc Sci 8: 169–194. https://doi.org/10.32327/ijmess/8.3.2019.11 doi: 10.32327/ijmess/8.3.2019.11
[51]	Kilian L (2008) Exogenous oil supply shocks: How big are they and how much do they matter for the U.S. economy? Rev Econ Stat 90: 216–240. https://doi.org/10.1162/rest.90.2.216 doi: 10.1162/rest.90.2.216
[52]	Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46: 33. https://doi.org/10.2307/1913643 doi: 10.2307/1913643
[53]	Liu C, Naeem MA, Rehman MU, et al. (2020) Oil as hedge, safe-haven, and diversifier for conventional currencies. Energies 13: 1–19. https://doi.org/10.3390/en13174354 doi: 10.3390/en13174354
[54]	Lo AW (2004) The adaptive markets hypothesis. J Portf Manage 30: 15–29. https://doi.org/10.3905/jpm.2004.442611 doi: 10.3905/jpm.2004.442611
[55]	Mensi W, Hammoudeh S, Reboredo JC, et al. (2014) Do global factors impact BRICS stock markets? A quantile regression approach (No. 159). https://doi.org/10.1016/j.ememar.2014.04.002
[56]	Mensi W, Nekhili R, Vo XV, et al. (2021) Oil and precious metals: Volatility transmission, hedging, and safe haven analysis from the Asian crisis to the COVID-19 crisis. Econ Anal Policy 71: 73–96. https://doi.org/10.1016/j.eap.2021.04.009 doi: 10.1016/j.eap.2021.04.009
[57]	Mohanty SK, Nandha M, Turkistani AQ, et al. (2011) Oil price movements and stock market returns: Evidence from Gulf Cooperation Council (GCC) countries. Glo Financ J 22: 42–55. https://doi.org/10.1016/j.gfj.2011.05.004 doi: 10.1016/j.gfj.2011.05.004
[58]	Mosteller Frederick, Tukey JW (1977) Data analysis and regression: A second course in statistics, In: Addison-Wesley series in behavioral science, Boston: Addison-Wesley Publishing Company.
[59]	Müller UA, Dacorogna MM, Davé RD, et al. (1997) Volatilities of different time resolutions—analyzing the dynamics of market components. J Empir Financ 4: 213–239. https://doi.org/10.1016/S0927-5398(97)00007-8 doi: 10.1016/S0927-5398(97)00007-8
[60]	Musialkowska I, Kliber A, Świerczyńska K, et al. (2020) Looking for a safe-haven in a crisis-driven Venezuela: The Caracas stock exchange vs gold, oil and bitcoin. Transforming Govern People Process Policy 14: 475–494. https://doi.org/10.1108/TG-01-2020-0009 doi: 10.1108/TG-01-2020-0009
[61]	Naifar N (2016) Do global risk factors and macroeconomic conditions affect global Islamic index dynamics? A quantile regression approach. Q Rev Econ Financ 61: 29–39. https://doi.org/10.1016/j.qref.2015.10.004 doi: 10.1016/j.qref.2015.10.004
[62]	Narayan PK, Gong Q, Ali Ahmed HJ (2021) Is there a pattern in how COVID-19 has affected Australia's stock returns? Appl Econ Lett 29: 179–182. https://doi.org/10.1080/13504851.2020.1861190 doi: 10.1080/13504851.2020.1861190
[63]	Nusair SA (2016) The effects of oil price shocks on the economies of the Gulf Co-operation Council countries: Nonlinear analysis. Energ Policy 91: 256–267. https://doi.org/10.1016/j.enpol.2016.01.013 doi: 10.1016/j.enpol.2016.01.013
[64]	Nusair SA, Al-Khasawneh JA (2018) Oil price shocks and stock market returns of the GCC countries: empirical evidence from quantile regression analysis. Econ Change Restructuring 51: 339–372. https://doi.org/10.1007/s10644-017-9207-4 doi: 10.1007/s10644-017-9207-4
[65]	OECD (2020) Global financial markets policy responses to COVID-19. Retrieved July 5, 2021. Available from: https://www.oecd.org/coronavirus/policy-responses/global-financial-markets-policy-responses-to-covid-19-2d98c7e0/.
[66]	Owusu Junior P, Adam AM, Tweneboah G (2020a) Connectedness of cryptocurrencies and gold returns: Evidence from frequency-dependent quantile regressions. Cogent Econ Financ 8: 1804037. https://doi.org/10.1080/23322039.2020.1804037 doi: 10.1080/23322039.2020.1804037
[67]	Owusu Junior P, Tiwari AK, Padhan H, et al. (2020b) Analysis of EEMD-based quantile-in-quantile approach on spot- futures prices of energy and precious metals in India. Resour Policy 68: 101731. https://doi.org/10.1016/j.resourpol.2020.101731 doi: 10.1016/j.resourpol.2020.101731
[68]	Owusu Junior P, Tweneboah G (2020) Are there asymmetric linkages between African stocks and exchange rates? Res Int Bus Financ 54. https://doi.org/10.1016/j.ribaf.2020.101245 doi: 10.1016/j.ribaf.2020.101245
[69]	Phillips PCB, Perron P (1988) Testing for a unit root in time series regression. Biometrika 75: 335–346. https://doi.org/10.1093/biomet/75.2.335 doi: 10.1093/biomet/75.2.335
[70]	Selmi R, Mensi W, Hammoudeh S, et al. (2018) Is Bitcoin a hedge, a safe haven or a diversifier for oil price movements? A comparison with gold. Energ Econ 74: 787–801. https://doi.org/10.1016/j.eneco.2018.07.007 doi: 10.1016/j.eneco.2018.07.007
[71]	Shahzad SJH, Bouri E, Roubaud D, et al. (2020) Safe haven, hedge and diversification for G7 stock markets: Gold versus bitcoin. Econ Mod 87: 212–224. https://doi.org/10.1016/j.econmod.2019.07.023 doi: 10.1016/j.econmod.2019.07.023
[72]	Siahaan GED, Robiyanto R (2021) Bond as a safe haven during market crash: Examination of COVID-19 pandemic in ASEAN-5. J Manage Entr 23: 1–9. https://doi.org/10.9744/jmk.23.1.1-9 doi: 10.9744/jmk.23.1.1-9
[73]	Smyth R, Narayan PK (2018) What do we know about oil prices and stock returns? Int Rev Financ Anal 57: 148–156. https://doi.org/10.1016/j.irfa.2018.03.010 doi: 10.1016/j.irfa.2018.03.010
[74]	Tavor T, Teitler-Regev S (2019) The impact of disasters and terrorism on the stock market. Jàmbá J Disaster Risk Stud 11. https://doi.org/10.4102/jamba.v11i1.534 doi: 10.4102/jamba.v11i1.534
[75]	Tweneboah G, Owusu Junior P, Oseifuah EK (2019) Integration of major African stock markets: Evidence from multi-scale wavelets correlation. Aca Acc Financ Std J 23: 1–15.
[76]	Tzeremes P (2021) Oil volatility index and Chinese stock markets during financial crisis: a time-varying perspective. J Chinese Econ Frgn Trd Std 14: 187–201. https://doi.org/10.1108/JCEFTS-08-2020-0051 doi: 10.1108/JCEFTS-08-2020-0051
[77]	Urquhart A, Zhang H (2019) Is Bitcoin a hedge or safe haven for currencies? An intraday analysis. Inl Rev of Financ Anal 63: 49–57. https://doi.org/10.1016/j.irfa.2019.02.009 doi: 10.1016/j.irfa.2019.02.009
[78]	Wu W, Lee CC, Xing W, et al. (2021) The impact of the COVID-19 outbreak on Chinese-listed tourism stocks. Financ Innov 7. https://doi.org/10.1186/s40854-021-00240-6 doi: 10.1186/s40854-021-00240-6
[79]	You W, Guo Y, Zhu HM, et al. (2017) Oil price shocks, economic policy uncertainty and industry stock returns in China: Asymmetric effects with quantile regression. Energ Econ 68: 1–18. https://doi.org/10.1016/j.eneco.2017.09.007 doi: 10.1016/j.eneco.2017.09.007
[80]	Kösedağlı BY, Kışla GH, Çatık AN (2021) The time-varying effects of oil prices on oil–gas stock returns of the fragile five countries. Financ Innov 7: 1–22. https://doi.org/10.1186/s40854-020-00224-y doi: 10.1186/s40854-020-00224-y
[81]	Zhu HM, Guo Y, You W, et al. (2016) The heterogeneity dependence between crude oil price changes and industry stock market returns in China: Evidence from a quantile regression approach. Energ Econ 55: 30–41. https://doi.org/10.1016/j.eneco.2015.12.027eco.2015.12.027 doi: 10.1016/j.eneco.2015.12.027eco.2015.12.027

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Quantitative Finance and Economics

Assessing the safe haven properties of oil in African stock markets amid the COVID-19 pandemic: a quantile regression analysis

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1. Introduction

2. Preliminaries

3. Solvability conditions and expression of general solution for system (1.1)

4. The solvability conditions and the general solutions to the Systems (1.2)–(1.4)

5. An application of the system (1.1)

6. Conclusions

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Quantitative Finance and Economics

Assessing the safe haven properties of oil in African stock markets amid the COVID-19 pandemic: a quantile regression analysis

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Abstract

1. Introduction

2. Preliminaries

3. Solvability conditions and expression of general solution for system (1.1)

4. The solvability conditions and the general solutions to the Systems (1.2)–(1.4)

5. An application of the system (1.1)

6. Conclusions

Author contributions

Acknowledgments

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