Demonstrating Invariant Encoding of Shapes Using A Matching Judgment Protocol

Ernest Greene; Michael J. Hautus; Ernest Greene; Michael J. Hautus

doi:10.3934/Neuroscience.2017.3.120

AIMS Neuroscience

2017, Volume 4, Issue 3: 120-147. doi: 10.3934/Neuroscience.2017.3.120

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Research article

Demonstrating Invariant Encoding of Shapes Using A Matching Judgment Protocol

Ernest Greene ^{1
,
,},
Michael J. Hautus ²

1.
Department of Psychology, University of Southern California, Los Angeles, CA 90089, USA
2.
School of Psychology, The University of Auckland, Auckland 1010, New Zealand

Received: 01 June 2017 Accepted: 18 August 2017 Published: 22 September 2017

Many theories have been offered to explain how the visual system registers, encodes, and recognizes the shape of an object. Some of the most influential assume that border lines and edges activate neurons in primary visual cortex, and these neurons encode the orientation, curvature, and linear extent of the shape as elemental cues. The present work challenges that assumption by showing that well-spaced dots can serve as effective shape cues. The experimental tasks drew on an inventory of unknown two-dimensional shapes, each being constructed as dots that marked the outer boundary, like an outline contour. A given shape was randomly picked from the inventory and was displayed only once as a target. The target shape was quickly followed by a low-density comparison shape that was derived from the target (matching) or from a different shape (non-matching). The respondent’s task was to provide a matching judgment, i.e., deciding whether the comparison shape was the “same” or “different.” Clear evidence of non-chance decisions was found even when the matching shapes displayed only 5% of the number of dots in the target shapes. Visual encoding mechanisms allow a shape to be identified when it is displayed at various locations on the retina, or with rotation or changes in size. A number of hierarchical network (connectionist) models have been developed to accomplish this encoding step, and these models appear especially credible because they are inspired by the anatomy and physiology of the visual system. The present work demonstrates above-chance translation, rotation, and size invariance for unknown shapes that were seen only once. This is clearly at odds with connectionist models that require extensive training before a shape can be identified irrespective of location, rotation, or size.

Keywords:

Citation: Ernest Greene, Michael J. Hautus. Demonstrating Invariant Encoding of Shapes Using A Matching Judgment Protocol[J]. AIMS Neuroscience, 2017, 4(3): 120-147. doi: 10.3934/Neuroscience.2017.3.120

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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]	Kohler W (1938) Gestaltprobleme und Anfänge einer Gestalttheorie. Jahresbericht über der Gesellshaft Physiologie. Translated by: Ellis WD, A Source Book of Gestalt Psychology, London: Routledge & Kegan Paul: 58.
[2]	Goldmeier E (1936/1972) Similarity in visually perceived forms. Psychol Issues 8: 1-135.
[3]	Greene E (2007) Recognition of objects displayed with incomplete sets of discrete boundary dots. Percept Mot Skills 104: 1043-1059.
[4]	Greene E, Visani A (2015) Recognition of letters displayed as briefly flashed dot patterns. Atten Percept Psychophys 77: 1955-1969. doi: 10.3758/s13414-015-0913-6
[5]	Greene E (2016) Information persistence evaluated with low-density dot patterns. Acta Psychol 170: 215-225.
[6]	Green DM, Swets JA (1966) Signal Detection Theory and Psychophysics. New York; Wiley.
[7]	Hautus MJ, van Hout D, Lee HS (2009) Variants of A Not-A and 2AFC tests: Signal Detection Theory models. Food Qual Prefer 20: 222-229. doi: 10.1016/j.foodqual.2008.10.002
[8]	Macmillan NA, Creelman CD (2005) Detection Theory: A User's Guide, New Jersey: Lawrence Erlbaum.
[9]	Hautus MJ (1995) Corrections for extreme proportions and their biasing effects on estimated values of d´. Behav Res Methods Instrum Comput 27: 46-51. doi: 10.3758/BF03203619
[10]	Miller J (1996) The sampling distribution of d'. Percept Psychophys 58: 65-72. doi: 10.3758/BF03205476
[11]	Hautus J (2012) SDT Assistant (version 1.0) [Software]. Available from http://hautus.org.
[12]	Hautus MJ (1997) Calculating estimates of sensitivity from group data: pooled versus averaged estimators. Behav Res Methods Instrum Comput 29: 556-562.
[13]	Hubel DH, Wiesel TN (1959) Receptive fields of single neurons in the cat's striate cortex. J Physiol 148: 574-591.
[14]	Hubel DH, Wiesel TN (1968) Receptive fields and functional architecture of monkey striate cortex. J Physiol 195: 215-243.
[15]	Selfridge OG (1959) Pandemonium: A Paradigm for Learning in the Mechanization of Thought Process, London; HM Stationary Office.
[16]	Sutherland NS (1968) Outlines of a theory of visual pattern recognition in animals and man. Proc R Soc Lond B Biol Sci 171: 297-317. doi: 10.1098/rspb.1968.0072
[17]	Binford TO (1971) Visual perception by computer. Proc IEEE Conf Syst Control, Miami, FL.
[18]	Barlow HB (1972) Single units and sensation: a neuron doctrine for perceptual psychology. Perception 1: 371-394. doi: 10.1068/p010371
[19]	Milner PM (1974) A model for visual shape recognition. Psychol Rev 81: 521-535.
[20]	Palmer SE (1975) Visual perception and world knowledge: notes on a model of sensory-cognitive interaction. In: Norman DA, Rumelhart DE eds, Explorations in Cognitions, San Francisco: WH Freeman & Co.: 279-307.
[21]	Marr D (1982) Vision: A Computational Investigation into the Human Representation and Processing of Information, New York: Freemen: 51-79.
[22]	DeValois RL, Devalois KK (1991) Vernier acuity with stationary moving Gabors. Vision Res 31: 1619-1626.
[23]	Panda R, Chatterji BN (1996) Gabor function: an efficient tool for digital image processing. IETE Tech Rev 13: 225-231. doi: 10.1080/02564602.1996.11416611
[24]	Kohonen T, Oja E (1998) Visual feature analysis by the self-organizing maps. Neural Comput Appl 7: 273-286.
[25]	Pettet MW, McKee SP, Grzywacz NM (1998) Constraints on long range interactions mediating contour detection. Vision Res 38: 865-879. doi: 10.1016/S0042-6989(97)00238-1
[26]	Pennefather PM, Chandna A, Kovacs I, et al. (1999) Contour detection threshold: repeatabililty and learning with 'contour cards.' Spat Vis 12: 257-266. doi: 10.1163/156856899X00157
[27]	Taylor G, Hipp D, Moser A, et al. (2014) The development of contour processing: evidence from physiology and psychophysics. Front Psychol 5: e719
[28]	Dong X, Chantlier MJ (2016) Perceptually motivated image features using contours. IEEE Trans Image Process 25: 5050-5062.
[29]	Edelman S (1999) Representation and Recognition in Vision, MIT Press.
[30]	Cooke T, Jakel F, Wallraven C, et al. (2007) Multimodal similarity and categorization of novel, three-dimensional objects. Neuropsychologia 45: 484-495. doi: 10.1016/j.neuropsychologia.2006.02.009
[31]	Hayworth KJ (2012) Dynamically partitionable autoassociative networks as a solution to the neural binding problem. Front Comput Neurosci 6: e73.
[32]	Rodriguez-Sanchez AJ, Tsotsos JK (2012) The roles of endstopped and curvature tuned computations in a hierarchical representation of 2D shape. PLoS ONE 7: e42058. doi: 10.1371/journal.pone.0042058
[33]	Hopfield JJ (1995) Pattern recognition computation using action potential timing for stimulus representation. Nature 376: 33-36.
[34]	Hopfield JJ (1996) Transforming neural computations and representing time. Proc Nat Acad Sci 83: 15440-15444.
[35]	Maass W (1997) Fast sigmoidal networks via spiking neurons. Neural Comput 9: 279-304. doi: 10.1162/neco.1997.9.2.279
[36]	McClelland JL (2013) Integrating probabilistic models of perception and interactive neural networks: a historical and tutorial review. Front Psychol 4: e503.
[37]	Guan T, Wang Y, Duan L, et al. (2015) On-device mobile landmark recognition using binarized descriptor and multifeature fusion. ACM Trans Intell Syst Technol 7: e12.
[38]	Wei B, Guan T, Duan L, et al. (2015) Wide area localization and tracking on camera phones for mobile segmented reality systems. Multimed Syst 21: 381-399. doi: 10.1007/s00530-014-0364-2
[39]	Pan H, Guan T, Luo Y, et al. (2016) Dense 3D reconstruction combining depth and RGB information. Neurocomputing 175: 644-651.
[40]	Ullman S (1976) Filling-in the gaps: the shape of subjective contours and a model for their generation. Biol Cybern 25: 1-6.
[41]	Sha'ashua A, Ullman S (1988) Structural saliency: the detection of globally salient structures using a locally connected network. In: Proc 2nd Intern Conf Comput Vision, Clearwater FL: 321-327.
[42]	Kellman PJ, Shipley TF (1991) A theory of visual interpolation in object rperception. Cogn Psychol 23: 141-221. doi: 10.1016/0010-0285(91)90009-D
[43]	Shipley TF, Kellman PJ (1992) Strength of visual interpolation depends on the ratio of physically specified to total edge length. Percept Psychophys 52: 97-106.
[44]	Cormen TH, Leherson CE, Rivest RL, et al. (2001) Single-source shortest paths and all pairs shortest paths. In: Introduction to Algorithms, MIT Press & McGraw-Hill: 580-642.
[45]	Field DJ, Hayes A, Hess RF (1993) Contour integration by the human visual system: evidence for a local "association field." Vision Res 33: 173-193.
[46]	Kwon TK, Agrawal K, Li Y, et al. (2016) Spatially-global integration of closed, fragmented contours by finding the shortest-path in a log-polar representation. Vision Res 126: 143-163.
[47]	Sceniak MP, Hawken MJ, Shapley R (2001) Visual spatial characterization of Macaque V1 neurons. J Neurophysiol 85: 1873-1887.
[48]	Bowers JS (2009) On the biological plausibility of grandmother cells: implications for neural network theories in psychology and neuroscience. Psychol Rev 116: 220-251.
[49]	Greene E (2008) Additional evidence that contour attributes are not essential cues for object recognition. Behav Brain Funct 4: e26.
[50]	Greene E (2016) How do we know whether three dots form an equilateral triangle? JSM Brain Sci 1: 1002.
[51]	Lichtsteiner P, Posch C, Delbruck T (2008) A 128x128 120 dB 15 µs latency asynchronous temporal contrast vision sensor. IEEE J Solid-State Circuits 43: 566-576. doi: 10.1109/JSSC.2007.914337
[52]	Robinson DA (1964) The mechanisms of human saccadic eye movement. J Physiol 174: 245-264.
[53]	Zimmermann E, Lappe M (2016) Visual space constructed by saccadic motor maps. Front Human Neurosci 10: e225.
[54]	McSorlely E, McCloy R, Williams L (2016) The concurrent programming of saccades. PLoS ONE 11: e0168724. doi: 10.1371/journal.pone.0168724
[55]	Bhutani N, Sengupta S, Basu D, et al. (2017) Parallel activation of prospective motor plans during visually-guided sequential saccades. Neurosci 45: 631-642.
[56]	Feldman JA, Ballard DH (1982) Connectionist models and their properties. Cogn Sci 6: 205-254. doi: 10.1207/s15516709cog0603_1
[57]	Fukushima K (1980) Neocognitron: a self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position. Biol Cybern 36: 193-202. doi: 10.1007/BF00344251
[58]	Rolls ET (1992) Neurophysiological mechanisms underlying face processing within and beyond the temporal cortical visual areas. Phil Trans R Soc 335: 11-21. doi: 10.1098/rstb.1992.0002
[59]	Wallis G, Rolls ET (1997) Invariant face and object recognition in the visual system. Prog Neurobiol 51: 167-194.
[60]	Riesenhuber M, Poggio T (2000) Models of object recognition. Nature Neurosci (S3): 1199-1204.
[61]	Pasupathy A, Connor CE (2001) Shape representation in area V4: position-specific tuning for boundary conformation. J Neurophysiol 86: 2505-2519.
[62]	Suzuki N, Hashimoto N, Kashimori Y, et al. (2004) A neural model of predictive recognition in form pathway of visual cortex. BioSystems 76: 33-42. doi: 10.1016/j.biosystems.2004.05.004
[63]	Pinto N, Cox DD, DeCarlo JJ (2008) Why is real-world visual object recognition hard? PLoS Comput Biol 4: e27. doi: 10.1371/journal.pcbi.0040027
[64]	Hancock PJB, Walton L, Mitchell G, et al. (2008) Segregation by onset asynchrony. J Vision 8: 1-21.
[65]	Karplus I, Goren M, Algorn D (1982) A preliminary experimental analysis of predator face recognition by Chromis caenuleus (Pisces, Pomacentridae). Z Tierpsychol 58: 53-65.
[66]	Siebeck UE, Parker AN, Sprenger D, et al. (2010) A species of reef fish that uses untraviolet patterns for covert face recognition. Curr Biol 20: 407-410.
[67]	Karplus I, Katzenstein R, Goren M (2006) Predator recognition and social facilitation of predator avoidance in coral reef fish Dascyllus marginatus juveniles. Mar Ecol Prog Ser 319: 215-223.
[68]	Siebeck UE, Litherland L, Wallis GM (2009) Shape learning and discrimination in reef fish. J Exp Biol 212: 2113-2119.
[69]	Newport C, Wallis G, Reshitnyk Y, et al. (2016) Discrimination of human faces by archerfish (Toxotes catareus). Sci Rep 6: e27523. doi: 10.1038/srep27523
[70]	Greschner M, Field GD, Li PH, et al. (2014) A polyaxonal amacrine cell population in the primate retina. J Neurosci 34: 3597-3606. doi: 10.1523/JNEUROSCI.3359-13.2014
[71]	Greschner M, Heitman AK, Field GD, et al. (2016) Identification of a retinal circuit for recurrent suppression using indirect electrical imaging. Curr Biol 26: 1935-1942. doi: 10.1016/j.cub.2016.05.051
[72]	Greene E (2007) Retinal encoding of ultrabrief shape recognition cues. PLoS One 2: e871.
[73]	Greene E (2016) Retinal encoding of shape boundaries. JSM Anat Physiol 1: 1002.

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