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Review Topical Sections

Biomedical diagnosis perspective of epigenetic detections using alpha-hemolysin nanopore

  • Received: 28 September 2015 Accepted: 02 November 2015 Published: 16 December 2015
  • The α-hemolysin nanopore has been studied for applications in DNA sequencing, various single-molecule detections, biomolecular interactions, and biochips. The detection of single molecules in a clinical setting could dramatically improve cancer detection and diagnosis as well as develop personalized medicine practices for patients. This brief review shortly presents the current solid state and protein nanopore platforms and their applications like biosensing and sequencing. We then elaborate on various epigenetic detections (like microRNA, G-quadruplex, DNA damages, DNA modifications) with the most widely used alpha-hemolysin pore from a biomedical diagnosis perspective. In these detections, a nanopore electrical current signature was generated by the interaction of a target with the pore. The signature often was evidenced by the difference in the event duration, current level, or both of them. An ideal signature would provide obvious differences in the nanopore signals between the target and the background molecules. The development of cancer biomarker detection techniques and nanopore devices have the potential to advance clinical research and resolve health problems. However, several challenges arise in applying nanopore devices to clinical studies, including super low physiological concentrations of biomarkers resulting in low sensitivity, complex biological sample contents resulting in false signals, and fast translocating speed through the pore resulting in poor detections. These issues and possible solutions are discussed.

    Citation: Yong Wang, Li-qun Gu. Biomedical diagnosis perspective of epigenetic detections using alpha-hemolysin nanopore[J]. AIMS Materials Science, 2015, 2(4): 448-472. doi: 10.3934/matersci.2015.4.448

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  • The α-hemolysin nanopore has been studied for applications in DNA sequencing, various single-molecule detections, biomolecular interactions, and biochips. The detection of single molecules in a clinical setting could dramatically improve cancer detection and diagnosis as well as develop personalized medicine practices for patients. This brief review shortly presents the current solid state and protein nanopore platforms and their applications like biosensing and sequencing. We then elaborate on various epigenetic detections (like microRNA, G-quadruplex, DNA damages, DNA modifications) with the most widely used alpha-hemolysin pore from a biomedical diagnosis perspective. In these detections, a nanopore electrical current signature was generated by the interaction of a target with the pore. The signature often was evidenced by the difference in the event duration, current level, or both of them. An ideal signature would provide obvious differences in the nanopore signals between the target and the background molecules. The development of cancer biomarker detection techniques and nanopore devices have the potential to advance clinical research and resolve health problems. However, several challenges arise in applying nanopore devices to clinical studies, including super low physiological concentrations of biomarkers resulting in low sensitivity, complex biological sample contents resulting in false signals, and fast translocating speed through the pore resulting in poor detections. These issues and possible solutions are discussed.


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

    $ {A11X1=B11,C11X1D11=E11,X2A22=B22,C22X2D22=E22,F11X1H11+X2F22=G11, $ (1.1)
    $ {A11X1=B11,C11X1D11=E11,X2A22=B22,C22X2D22=E22,F11X1+H11X2F22=G11, $ (1.2)
    $ {A11X1=B11,C11X1D11=E11,A22X2=B22,C22X2D22=E22,F11X1+H11X2F22=G11, $ (1.3)
    $ {A11X1=B11,C11X1D11=E11,A22X2=B22,C22X2D22=E22,F11X1+X2F22=G11, $ (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.

    $ H={v0+v1i+v2j+v3k|i2=j2=k2=ijk=1,v0,v1,v2,v3R}. $

    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

    $ AXA=A,XAX=X,(AX)=AX,(XA)=XA, $

    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:

    $ {A1V=C1, VB1=C2,A3X+YB3=C3,A2Y+ZB2+A5VB5=C5,A4W+ZB4=C4,{A1V=C1, VB1=C2,A3X+YB3=C3,A2Z+YB2+A5VB5=C5,A4Z+WB4=C4,{A1V=C1, VB1=C2,A3X+YB3=C3,A2Y+ZB2+A5VB5=C5,A4Z+WB4=C4. $ (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:

    $ {A1X+YB1=C1,A2Y+ZB2=C2,A3W+ZB3=C3,{A1X+YB1=C1,A2Z+YB2=C2,A3Z+WB3=C3,{A1X+YB1=C1,A2Y+ZB2=C2,A3Z+WB3=C3. $ (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):

    $ A1U+VB1+A2XB2+A3YB3+A4ZB4=B. $ (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:

    $ {A11X1=B11,C11X1D11=E11,X2A22=B22,C22X2D22=E22,F11X1+X2F22=G11. $ (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:

    $ {A11X1=B11,C11X1Cη11=E11,F11X1Fη11+(F22X1)η=G11 $ (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.

    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(ABLDREC0) = r(AB0C0E0D0)-r(D)-r(E). $

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

    $ (1)(Aη)=(A)η,(Aη)=(A)η;(2)r(A)=r(Aη)=r(Aη);(3)(LA)η=η(LA)η=(LA)η=LAη=RAη,(4)(RA)η=η(RA)η=(RA)η=RAη=LAη;(5)(AA)η=(A)ηAη=(AA)η=Aη(A)η;(6)(AA)η=Aη(A)η=(AA)η=(A)ηAη. $

    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

    $ AXB=C $

    is consistent if, and only if,

    $ RAC=0,CLB=0. $

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

    $ X=ACB+LAU+VRB, $

    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.

    $ C1X1+X2D1+C2Y1D2+C3Y2D3+C4Y3D4=Z. $ (2.1)

    Denote

    $ RC1C2=C12,RC1C3=C13,RC1C4=C14,D2LD1=D21,D31LD21=N32,D3LD1=D31,D4LD1=D41,RC12C13=M23,S12=C13LM23,RC1ZLD1=T1,C32=RM23RC12,A1=C32C14,A2=RC12C14,A3=RC13C14,A4=C14,D13=LD21LN32,B1=D41,B2=D41LD31,B3=D41LD21,B4=D41D13,E1=C32T1,E2=RC12T1LD31,E3=RC13T1LD21,E4=T1D13,A24=(LA2,LA4),B13=(RB1RB3),A11=LA1,B22=RB2,A33=LA3,B44=RB4,E11=RA24A11,E22=RA24A33,E33=B22LB13,E44=B44LB13,N=RE11E22,M=E44LE33,K=K2K1,E=RA24KLB13,S=E22LN,K11=A2LA1,G1=E2A2A1E1B1B2,K22=A4LA3,G2=E4A4A3E3B3B4,K1=A1E1B1+LA1A2E2B2,K2=A3E3B3+LA3A4E4B4. $

    Then, the following statements are equivalent:

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

    $ \mathrm{(2)} $

    $ RAiEi=0,EiLBi=0(i=¯1,4),RE11ELE44=0. $

    $ \mathrm{(3)} $

    $ r(ZC2C3C4C1D10000)=r(D1)+r(C2,C3,C4,C1),r(ZC2C4C1D3000D1000)=r(C2,C4,C1)+r(D3D1),r(ZC3C4C1D2000D1000)=r(C3,C4,C1)+r(D2D1),r(ZC4C1D200D300D100)=r(D2D3D1)+r(C4,C1),r(ZC2C3C1D4000D1000)=r(C2,C3,C1)+r(D4D1),r(ZC2C1D300D400D100)=r(D3D4D1)+r(C2,C1),r(ZC3C1D200D400D100)=r(D2D4D1)+r(C3,C1),r(ZC1D20D30D40D10)=r(D2D3D4D1)+r(C1),r(ZC2C1000C4D3000000D1000000000ZC3C1C4000D2000000D1000D400D4000)=r(D30D100D20D1D4D4)+r(C2C100C400C3C1C4). $

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

    $ X1=C1(ZC2Y1D2C3Y2D3C4Y3D4)C1U1D1+LC1U2,X2=RC1(ZC2Y1D2C3Y2D3C4Y3D4)D1+C1C1U1+U3RD1,Y1=C12TD21C12C13M23TD21C12S12C13TN32D31D21C12S12U4RN32D31D21+LC12U5+U6RD21,Y2=M23TD31+S12S12C13TN32+LM23LS12U7+U8RD31+LM23U4RN32,Y3=K1+LA2V1+V2RB1+LA1V3RB2, or Y3=K2LA4W1W2RB3LA3W3RB4, $

    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} $,

    $ V1=(Im,0)[A24(KA11V3B22A33W3B44)A24U11B13+LA24U12],W1=(0,Im)[A24(KA11V3B22A33W3B44)A24U11B13+LA24U12],W2=[RA24(KA11V3B22A33W3B44)B13+A24A24U11+U21RB13](0In),V2=[RA24(KA11V3B22A33W3B44)B13+A24A24U11+U21RB13](In0),V3=E11KE33E11E22NKE33E11SE22KME44E33E11SU31RME44E33+LE11U32+U33RE33,W3=NKE44+SSE22KM+LNLSU41+LNU31RMU42RE44, $

    $ 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.

    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

    $ {A1=C11LA11,P1=E11C11A11B11D11,B2=RA22D22,P2=E22C22B22A22D22,^B1=RB2RA22F22,^A2=F11LA11LA1,^A3=F11LA11,^B3=RD11H11,^A4=LC22,^B4=RA22F22,H11L^B1=^B11,P=G11F11A11B11H11F11LA11A1P1D11H11B22A22F22C22P2B2RA22F22, $ (3.1)
    $ {^B22L^B11=N1,^B3L^B1=^B22,^B4L^B1=^B33,R^A2^A3=^M1,S1=^A3L^M1,T1=PL^B1,C=R^M1R^A2,C1=C^A4,C2=R^A2^A4,C3=R^A3^A4,C4=^A4,D=L^B11LN1,D1=^B33,D2=^B33L^B22,D4=^B33D,E1=CT1,E2=R^A2T1L^B22,E3=R^A3T1L^B11,E4=T1D,^C11=(LC2,LC4),D3=^B33L^B11,^D11=(RD1RD3),^C22=LC1,^D22=RD2,^C33=LC3,^D33=RD4,^E11=R^C11^C22,^E22=R^C11^C22,^E33=^D22L^D11,^E44=^D33L^D11,M=R^E11^E22,N=^E44L^E33,F=F2F1,E=R^C11FL^D11,S=^E22LM,^F11=C2LC1,G1=E2C2C1E1D1D2,^F22=C4LC3,G2=E4C4C3E3D3D4,F1=C1E1D1+LC1C2E2D2,F2=C3E3D3+LC3C4E4D4. $ (3.2)

    Then, the following statements are equivalent:

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

    $ \mathrm{(2)} $

    $ RA11B11=0,RA1P1=0,P1LD11=0,B22LA22=0,RC22P2=0,P2LB2=0,RCiEi=0,EiLDi=0(i=¯1,4),R^E11EL^E44=0. $

    $ \mathrm{(3)} $

    $ r(B11,A11)=r(A11),r(E11C11B11D11A11)=r(C11A11), $ (3.3)
    $ r(E11D11)=r(D11),r(B22A22)=r(A22), $ (3.4)
    $ r(E22,C22)=r(C22),r(E22C22B22D22A22)=r(D22,A22), $ (3.5)
    $ r(F220D22A22B11H11A1100C22G11C22F11E22C22B22)=r(F22,D22,A22)+r(A11C22F11), $ (3.6)
    $ r(H110D1100F2200D22A220C11E11000A11B11D1100C22G11C22F110E22C22B22)=r(C11A11C22F11)+r(H11D1100F220D22A22), $ (3.7)
    $ r(H11000F220D22A220A1100C22G11C22F11E22C22B22)=r(H1100F22D22A22)+r(A11C22F11), $ (3.8)
    $ r(H1100F22D22A22C22G11E22C22B22)=r(H1100F22D22A22), $ (3.9)
    $ r(G11F11B22F220A22B11H11A110)=r(F11A11)+r(F22,A22), $ (3.10)
    $ r(G11F110B22H110D110F2200A220C11E1100A11B11D110)=r(H11D110F220A22)+r(F11C11A11), $ (3.11)
    $ r(G11F11B22H1100F220A220A110)=r(H110F22A22)+r(F11A11), $ (3.12)
    $ r(G11B22H110F22A22)=r(H110F22A22), $ (3.13)
    $ r(H11000000D110F220000D22A220000H1100000000F22D22A220000F220F2200000A220C1100000E1100A1100000B11D110C22G11C22F11000E22C22B2200)=r(H1100000D110F22000D22A22000H110000000F22D22A220000F22F2200000A22)+r(C11A11C22F11). $ (3.14)

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

    $ A11X1=B11, X2A22=B22, $ (3.15)
    $ C11X1D11=E11, C22X2D22=E22, $ (3.16)

    and

    $ F11X1H11+X2F22=G11. $ (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,

    $ RA11B11=0, B22LA22=0. $ (3.18)

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

    $ X1=A11B11+LA11U1, X2=B22A22+U2RA22. $ (3.19)

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

    $ A1U1D11=P1, C22U2B2=P2, $ (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,

    $ RA1P1=0, P1LD11=0, RC22P2=0, P2LB2=0. $ (3.21)

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

    $ U1=A1P1D11+LA1W1+W2RD11,U2=C22P2B2+LC22W3+W4RB2. $ (3.22)

    Comparing (3.22) and (3.19), hence,

    $ X1=A11B11+LA11A1P1D11+LA11LA1W1+LA11W2RD11,X2=B22A22+C22P2B2RA22+LC22W3RA22+W4RB2RA22. $ (3.23)

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

    $ W4^B1+^A2W1H11+^A3W2^B3+^A4W3^B4=P, $ (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,

    $ RCiEi=0,EiLDi=0(i=¯1,4),R^E11EL^E44=0. $ (3.25)

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

    $ W1=^A2T^B11^A2^A3^M1T^B11^A2S1^A3TN1^B22^B11^A2S1V4RN1^B22^B11+L^A2V5+V6R^B11,W2=^M1T^B22+S1S1^A3TN1+L^M1LS1V7+V8R^B22+L^M1V4RN1,W3=F1+LC2^V1+^V2RD1+LC1^V3RD2, or W3=F2LC4V1V2RD3LC3V3RD4,W4=(P^A2W1H11^A3W2^B3^A4W3^B4)^B1+V3R^B1, $

    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} $,

    $ ^V1=(Im,0)[^C11(F^C22V3^D22^C33^V3^D33)^C11U11^D11+L^C11U12],V1=(0,Im)[^C11(F^C22V3^D22^C33^V3^D33)^C11U11^D11+L^C11U12],V2=[R^C11(F^C22V3^D22^C33^V3^D33)^D11+^C11^C11U11+U21R^D11](0In),^V2=[R^C11(F^C22V3^D22^C33^V3^D33)^D11+^C11^C11U11+U21R^D11](In0),^V3=^E11F^E33^E11^E22MF^E33^E11S^E22FN^E44^E33^E11SU31RN^E44^E33+L^E11U32+U33R^E33,V3=MF^E44+SS^E22FN+LMLSU41+LMU31RNU42R^E44, $

    $ 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

    $ A11X01=B11, X02A22=B22,A1U01D11=P1, C22U02B2=P2 $

    holds, where

    $ X01=A11B11,U01=A1P1D11,X02=B22A22,U02=C22P2B2, $

    $ 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

    $ (3.18)r(RA11B11)=0r(B11,A11)=r(A11)(3.3),(3.21)r(RA1P1)=0r(P1,A1)=r(A1)r(E11C11A11B11D11,C11LA11)=r(C11LA11)r(E11C11B11D11A11)=r(C11A11)(3.3),(3.21)r(P1LD11)=0r(P1D11)=r(D11)r(E11C11A11B11D11D11)=r(D11)r(E11D11)=r(D11)(3.4),(3.18)r(B22LA22)=0r(B22A22)=r(A22)(3.4). $

    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:

    $ r(P^A2^A3^A4^B1000)=r(^B1)+r(^A2,^A3,^A4), $ (3.26)
    $ r(P^A2^A4^B300^B100)=r(^A2,^A4)+r(^B3^B1), $ (3.27)
    $ r(P^A3^A4H1100^B100)=r(^A3,^A4)+r(H11^B1), $ (3.28)
    $ r(P^A4H110^B30^B10)=r(H11^B3^B1)+r(^A4), $ (3.29)
    $ r(P^A2^A3^B400^B100)=r(^A2,^A3)+r(^B4^B1), $ (3.30)
    $ r(P^A2^B30^B40^B10)=r(^B3^B4^B1)+r(^A2), $ (3.31)
    $ r(P^A3H110^B40^B10)=r(H11^B4^B1)+r(^A3), $ (3.32)
    $ r(PH11^B3^B4^B1)=r(H11^B3^B4^B1), $ (3.33)
    $ r(P^A200^A4^B30000^B1000000P^A3^A400H110000^B100^B40^B400)=r(^B30^B100H110^B1^B4^B4)+r(^A20^A40^A3^A4), $ (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

    $ (3.26)=r(P^A2^A3^A4^B1000)=r(^B1)+r(^A2,^A3,^A4)r(G11F11X01H11F11LA11U01H11X02F22U02RA22F22F11LA11LA1F11LA11LC22RB2RA22F22000)=r(RB2RA22F22)+r(F11LA11LA1,F11LA11,LC22)r(G11F11X01H11X02F22U02RA22F22F11I0RA22F2200B20A110000C220)=r(RA22F22,B2)+r(F11IA1100C22)r(G11F11IU02B20F2200B2A22B11H11A11000C22X02F220C2200)=r(F22,D22,A22)+r(F11IA1100C22)r(F220D22A22B11H11A1100C22G11C22F11E22C22B22)=r(F22,D22,A22)+r(A11F11C22)(3.6). $

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

    $ (3.30)=r(P^A2^A3^B400^B100)=r(^A2,^A3)+r(^B4^B1)r(G11F11X01H11F11LA11U01H11X02F22U02RA22F22F11LA11LA1F11LA1RA22F2200RB2RA22F2200)=r(F11LA11LA1,F11LA11)+r(RA22F22RB2RA22F22)r(G11F11X01H11F11B22F220A220A110)=r(F11A11)+r(F22,A22)r(G11F11B22F220A22B11H11A110)=r(F11A11)+r(F22,A22)(3.10). $

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

    $ (3.34)=r(P^A200^A4^B30000^B1000000P^A3^A400H110000^B100^B40^B400)=r(^B30^B100H110^B1^B4^B4)+r(^A20^A40^A3^A4)r(PF11LA11LA100LC22RD11H110000RB2RA22F22000000PF11LA11LC2200H110000RB2RA22F2200RA22F220RA22F2200)=r(RD11H110RB2RA22F2200H110RB2RA22F22RA22F22RA22F22)+r(F11LA11LA10LC220F11LA11LC22)r(PF11LA1100LC22000H110000D1100RA22F2200000B2000G11+X02F22+U02RA22F22F11LA11LC2200000H110000000RA22F220000B2RA22F220RA22F22000000A1000000)=r(H110D1100RA2200B200H110000RA22F2200B2RA22F22RA22F22000)+r(F11LA110LC220F11LA11LC22A100)r(H11000000D110F220000D22A220000H1100000000F22D22A220000F220F2200000A220C1100000E1100A1100000B11D110C22G11C22F11000E22C22B2200)=r(H1100000D110F22000D22A22000H110000000F22D22A220000F22F2200000A22)+r(C11A11C22F11)(3.14). $

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

    $ X1=A11B11+LA11A1P1D11+LA11LA1W1+LA11W2RD11,X2=B22A22+C22P2B2RA22+LC22W3RA22+W4RB2RA22, $

    where

    $ W1=^A2T^B11^A2^A3^M1T^B11^A2S1^A3TN1^B22^B11^A2S1V4RN1^B22^B11+L^A2V5+V6R^B11,W2=^M1T^B22+S1S1^A3TN1+L^M1LS1V7+V8R^B22+L^M1V4RN1,W3=F1+LC2^V1+^V2RD1+LC1^V3RD2, or W3=F2LC4V1V2RD3LC3V3RD4,W4=(P^A2W1H11^A3W2^B3^A4W3^B4)^B1+V3R^B1,^V1=(Im,0)[^C11(F^C22V3^D22^C33^V3^D33)^C11U11^D11+L^C11U12],V1=(0,Im)[^C11(F^C22V3^D22^C33^V3^D33)^C11U11^D11+L^C11U12],V2=[R^C11(F^C22V3^D22^C33^V3^D33)^D11+^C11^C11U11+U21R^D11](0In),^V2=[R^C11(F^C22V3^D22^C33^V3^D33)^D11+^C11^C11U11+U21R^D11](In0),^V3=^E11F^E33^E11^E22MF^E33^E11S^E22FN^E44^E33^E11SU31RN^E44^E33+L^E11U32+U33R^E33,V3=MF^E44+SS^E22FN+LMLSU41+LMU31RNU42R^E44, $

    $ 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

    $ T=C11LA11,K=RA22D22, B1=RKRA22F22,A1=F11LA11LT,C3=F11LA11,D3=RD11,C4=LC22,D4=RA22F22,Aα=RA1C3,Bβ=D3LB1,Cc=RAαC4,Dd=D4LB1,E=RA1E1LB1,A=A11B11+LA11T(E11C11A11B11D11)D,B=B22A22+C22(E22C22B22A22D22)KRA22,E1=G11F11ABF22,M=RAαCc,N=DdLBβ,S=CcLM. $

    Then, the following statements are equivalent:

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

    $ \mathrm{(2)} $

    $ RA11B11=0,B22LA22=0,RC22E22=0,E11LD11=0,RT(E11C11A11B11D11)=0,(E22C22B22A22D22)LK=0,RMRAαE=0,ELBβLN=0,RAαELDd=0,RCcELBβ=0. $

    $ \mathrm{(3)} $

    $ r(B11,A11)=r(A11),r(E11C11B11D11A11)=r(C11A11),r(E11D11)=r(D11),r(B22A22)=r(A22),r(E22,C22)=r(C22),r(E22C22B22D22A22)=r(D22,A22),r(F220D22A22B11A1100C22G11C22F11E22C22B22)=r(F22,D22,A22)+r(A11C22F11),r(0F22D11D22A22C11E1100A11B11D1100C22F11C22G11D11E22C22B22)=r(C11A11C22F11)+r(F22D11,D22,A22),r(G11F11B22F220A22B11A110)=r(F11A11)+r(F22,A22),r(F11G11D11B220F22D11A22C11E110A11B11D110)=r(F22D11,A22)+r(F11C11A11). $

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

    Example 3.4. Conside the matrix equation (1.1)

    $ A11=(a111a121),B11=(b111b112b121b122),C11=(c111c121),D11=(d111d121),E11=(e111e121),A22=(a211a212),B22=(b211b212b221b222),C22=(c211c212c221c222),D22=(d211),E22=(e211e221),F11=(f111f121),H11=(h111h112h121h122),F22=(f211f212),G11=(g111g112g121g122), $

    where

    $ a111=0.9787+0.5005i+0.0596j+0.0424k,a121=0.7127+0.4711i+0.6820j+0.0714k,b111=0.5216+0.8181i+0.7224j+0.6596k,b112=0.9730+0.8003i+0.4324j+0.0835k,b121=0.0967+0.8175i+0.1499j+0.5186k,b122=0.6490+0.4538i0.8253j+0.1332k,c111=0.1734+0.8314i+0.0605j+0.5269k,c121=0.3909+0.8034i+0.3993j+0.4168k,d111=0.6569+0.2920i+0.0159j+0.1671k,d121=0.6280+0.4317i+0.9841j+0.1062k,e111=0.3724+0.4897i+0.9516j+0.0527k,e121=0.1981+0.3395i+0.9203j+0.7379k,a211=0.2691+0.4228i+0.5479j+0.9427k,a212=0.4177+0.9831i+0.3015j+0.7011k,b211=0.6663+0.6981i+0.1781j+0.9991k,b212=0.0326+0.8819i+0.1904j+0.4607k,b221=0.5391+0.6665i+0.1280j+0.1711k,b222=0.5612+0.6692i+0.3689j+0.9816k,c211=0.1564+0.6448i+0.1909j+0.4820k,c212=0.5895+0.3846i+0.2518j+0.6171k,c221=0.8555+0.3763i+0.4283j+0.1206k,c222=0.2262+0.5830i+0.2904j+0.2653k,d211=0.8244+0.9827i+0.7302j+0.3439k,e211=0.5847+0.9063i+0.8178j+0.5944k,e221=0.1078+0.8797i+0.2607j+0.0225k,f111=0.4253+0.1615i+0.4229j+0.5985k,f121=0.3127+0.1788i+0.0942j+0.4709k,h111=0.6959+0.6385i+0.0688j+0.5309k,h112=0.4076+0.7184i+0.5313j+0.1056k,h121=0.6999+0.0336i+0.3196j+0.6544k,h122=0.8200+0.9686i+0.3251j+0.6110k,f211=0.7788+0.4235i+0.0908j+0.2665k,f212=0.1537+0.2810i+0.4401j+0.5271k,g111=0.4574+0.5181i+0.6377j+0.2407k,g112=0.2891+0.6951i+0.2548j+0.6678k,g121=0.8754+0.9436i+0.9577j+0.6761k,g122=0.6718+0.0680i+0.2240j+0.8444k. $

    Computing directly yields the following:

    $ r(B11A11)=r(A11)=2,r(E11C11B11D11A11)=r(C11A11)=2,r(E11D11)=r(D11)=1,r(B22A22)=r(A22)=2,r(E22C22)=r(C22)=2,r(E22C22B22D22A22)=r(D22A22)=3,r(F220D22A22B11H11A1100C22G11C22F11E22C22B22)=r(F22D22A22)+r(A11C22F11)=5,r(H110D1100F2200D22A220C11E11000A11B11D1100C22G11C22F110E22C22B22)=r(C11A11C22F11)+r(H11D1100F220D22A22)=7,r(H11000F220D22A220A1100C22G11C22F11E22C22B22)=r(H1100F22D22A22)+r(A11C22F11)=6,r(H1100F22D22A22C22G11E22C22B22)=r(H1100F22D22A22)=5,r(G11F11B22F220A22B11H11A110)=r(F11A11)+r(F22,A22)=5,r(G11F110B22H110D110F2200A220C11E1100A11B11D110)=r(H11D110F220A22)+r(F11C11A11)=6,r(G11F11B22H1100F220A220A110)=r(H110F22A22)+r(F11A11)=5, r(G11B22H110F22A22)=r(H110F22A22)=4,r(H11000000D110F220000D22A220000H1100000000F22D22A220000F220F2200000A220C1100000E1100A1100000B11D110C22G11C22F11000E22C22B2200)=r(H1100000D110F22000D22A22000H110000000F22D22A220000F22F2200000A22)+r(C11A11C22F11)=11. $

    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

    $ X1=(0.4946+0.1700i0.1182j0.3692k0.40510.0631i0.2403j0.1875k),X2=(0.0122+0.2540i0.3398j0.3918k0.70020.3481i0.2169j+0.0079k). $

    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

    $ A1=C11LA11,P1=E11C11A11B11D11,B2=RA22D22,P2=E22C22B22A22D22,^A1=F11LA11LA1,^A2=F11LA1,^B2=RD11,^A3=H11LC22,^B3=RA22F22,^B4=RB2RA22F22,B=G11F11A11B11F11LA11A1P1D11H11B22A22F22H11C22P2B2RA22F22,R^A1^A2=A12,R^A1^A3=A13,R^A1H11=A14,^B3L^B2=N1,RA12A13=M1,S1=A13LM1,R^A1B=T1,C=RM1RA12,^C1=CA14,^C2=RA12A14,^C3=RA13A14,^C4=A14,D=L^B2LN1,^D1=^B4,^D2=^B4L^B3,^D3=^B4L^B2,^D4=^B4D,^E1=CT1,^E2=RA12T1L^B3,^E3=RA13T1L^B2,^E4=T1D,C24=(L^C2,L^C4),D13=(R^D1R^D3),C12=L^C1,D12=R^D2,C33=L^C3,D33=R^D4,E24=RC24C12,E13=RC24C33,E33=D12LD13,E44=D33LD13,M=RE24E13,N=E44LE33,F=F2F1,E=RC24FLD13,S=E13LM,^F11=^C2L^C1,^G1=^E2^C2^C1^E1^D1^D2,F33=^C4L^C3,^G2=^E4^C4^C3^E3^D3^D4,F1=^C1^E1^D1+L^C1^C2^E2^D2,F2=^C3^E3^D3+L^C3^C4^E4^D4. $

    Then, the following statements are equivalent:

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

    $ \mathrm{(2)} $

    $ RA11B11=0,RA1P1=0,P1LD11=0,B22LA22=0,RC22P2=0,P2LB2=0,R^Ci^Ei=0,^EiL^Di=0(i=¯1,4),RE24ELE44=0. $

    $ \mathrm{(3)} $

    $ r(B11,A11)=r(A11),r(E11C11B11D11A11)=r(C11A11),r(E11D11)=r(D11),r(B22,A22)=r(A22),r(E22,C22)=r(C22),r(E22C22B22D22A22)=r(D22,A22),r(G11D11F11H11E11C110B11D11A110)=r(F11H11C110A110),r(G11D11F11H110F22D1100A22E11C1100B11D11A1100)=r(F22,A22)+r(F11H11C110A110),r(H11F11G11D110C11E110A11B11D11)=r(H11F110C110A11),r(H11F110G11D1100A22F22D110C110E110A110B11D11)=r(F22D11,A22)+r(H11F110C110A11),r(G11D11F11H1100F22D1100D22A22E11C1100000C22E22C22B22B11D11A11000)=r(F11H11C1100C22A110)+r(F22,D22,A22),r(G11D11F11H11B22F22D110A22E11C110B11D11A110)=r(F11C11A11)+r(F22,A22),r(H11F1100G11D1100D22A22F22D110C1100E110A1100B11D11C220E22C22B220)=r(H11F110C110A11C220)+r(D22,A22,F22D11),r(F11H11B22G11D110A22F22D11C110E11A110B11D11)=r(F11C11A11)+r(A22,F22D11),r(G11F1100H1100H5B220F22000000A22000H11F11H110H11B220G11D1100000D22A220F22D1100C2200E22000000C110000E11000A110000B11D11B11A110000000)=r(F2200A2200D22A220F22D11)+r(F1100H110H11F11H110C220000C11000A110A11000). $

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

    $ X1=A11B11+LA11A1P1D11+LA11LA1W1+LA11W2RD11,X2=B22A22+C22P2B2RA22+LC22W3RA22+W4RB2RA22, $

    where

    $ W1=^A1(B^A2W1^B2^A3W3^B3H11W4^B4)+L^A1U1,W2=A12T^B2A12A13M1T^B2A12S1A13TN1^B3^B2A12S1U2RN1^B3^B2+LA12U3+U4R^B2,W3=M1T^B3+S1S1A13TN1+LM1LS1U5+U6R^B3+LM1U2RN1,W4=F1+L^C2V1+V2R^D1+L^C1V3R^D2, or W4=F2L^C4^V1^V2R^D3L^C3^V3R^D4, $

    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} $,

    $ V1=(Im,0)[C24(FC12V3D12C33^V3D33)C24U11D13+LC24U12],^V1=(0,Im)[C24(FC12V3D12C33^V3D33)C24U11D13+LC24U12],^V2=[RC24(FC12V3D12C33^V3D33)D13+C24C24U11+U21RD13](0In),V2=[RC24(FC12V3D12C33^V3D33)D13+C24C24U11+U21RD13](In0),V3=E24FE33E24E13MFE33E24SE13FNE44E33E24SU31RNE44E33+LE24U32+U33RE33,^V3=MFE44+SSE13FN+LMLSU41+LMU31RNU42RE44, $

    $ 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

    $ A1=C11LA11,P1=E11C11A11B11D11,A2=C22LA22,P2=E22C22A22B22D22,^A1=F11LA11LA1,^A2=F11LA11,^B2=RD11,^A11=H11LA22LA2,^A22=H11LA22,^B4=RD22F22,B=G11F11A11B11F11LA11A1P1D11H11A22B22F22H11LA22A2P2D22F22,R^A1^A2=A12,R^A1^A11=A13,R^A1^A22=A33,F22L^B2=N1,RA12A13=M1,S1=A13LM1,R^A1B=T1,C=RM1RA12,^C1=CA33,^C2=RA12A33,^C11=RA13A33,^C22=A33,D=L^B2LN1,^D1=^B4,^D2=^B4LF22,^D11=^B4L^B2,^D22=^B4D,^E1=CT1,^E2=RA12T1LF22,^E11=RA13T1L^B2,^E4=T1D,C24=(L^C2,L^C22),D13=(R^D1R^D11),C21=L^C1,D12=R^D2,C33=L^C11,D33=R^D22,E11=RC24C21,E22=RC24C33,E33=D12LD13,E44=D33LD13,M=RE11E22,N=E44LE33,F=F2F1,E=RC24FLD13,S=E22LM,^F11=^C2L^C1,^G1=^E2^C2^C1^E1^D1^D2,^F22=^C22L^C11,^G2=^E4^C22^C11^E11^D11^D22,F1=^C1^E1^D1+L^C1^C2^E2^D2,F2=^C11^E11^D11+L^C11^C22^E4^D22. $

    Then, the following statements are equivalent:

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

    $ \mathrm{(2)} $

    $ RA11B11=0,RA1P1=0,P1LD11=0,RA22B22=0,RA2P2=0,P2LD22=0,R^Ci^Ei=0,R^C11^E11=0,R^C22^E4=0,^EiL^Di=0(i=¯1,2),^E11L^D11=0,^E4L^D22=0,RE11ELE44=0. $

    $ \mathrm{(3)} $

    $ r(B11,A11)=r(A11),r(E11C11B11D11A11)=r(C11A11),r(E11D11)=r(D11),r(B22,A22)=r(A22),r(E22C22B4D22A22)=r(C22A22),r(E22D22)=r(D22),r(G11F11H11B11A110B22F220A22)=r(F11H11A1100A22),r(G11F11H11F2200B11A11000A22)=r(F22)+r(F11H11A1100A22),r(H11F11G11D11A220B22F22D110C11E110A11B11D11)=r(H11F110C110A11A220),r(H11F11G11D1100F22D110C11E110A11B11D11A2200)=r(H11F110C110A11A220)+r(F22D11),r(G11F11H110F2200D22B11A110000C22E2200A22B22D22)=r(F11H11A1100C220A22)+r(F22,D22),r(G11F11F220B11A11)=r(F11A11)+r(F22),r(H11F110G11D1100D22F22D11C220E2200C110E11A2200B22F22D110A110B11D11)=r(H11F11C2200C11A2200A11)+r(D22,F22D11),r(F11G11D110F22D11C11E11A11B11D11)=r(F11C11A11)+r(F22D11),r(G11F11000H110F2200000000G11D11H11F11H11000F22D11000B22B11A1100000000C2200E2200E110C110000B22F22D11A2200000B11D110A110000000A220)=r(F22000D22F22D11)+r(F1100H110H11F11H110C22000A220000C11000A110A11000000A22). $

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

    $ X1=A11B11+LA11A1P1D11+LA11LA1W1+LA11W2RD11,X2=A22B4+LA22A2P2D22+LA22LA2W3+LA22W4RD22, $

    where

    $ W1=^A1(B^A2W1^B2^A11W3F22^A22W4^B4)+L^A1U1,W2=A12T^B2A12A13M1T^B2A12S1A13TN1F22^B2A12S1U2RN1F22^B2+LA12U3+U4R^B2,W3=M1TF22+S1S1A13TN1+LM1LS1U5+U6RF22+LM1U2RN1,W4=F1+L^C2V1+V2R^D1+L^C1V3R^D2, or W4=F2L^C22^V1^V2R^D11L^C11^V3R^D22, $

    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} $,

    $ V1=(Im,0)[C24(FC21V3D12C33^V3D33)C24U11D13+LC24U12],^V1=(0,Im)[C24(FC21V3D12C33^V3D33)C24U11D13+LC24U12],^V2=[RC24(FC21V3D12C33^V3D33)D13+C24C24U11+U21RD13](0In),V2=[RC24(FC21V3D12C33^V3D33)D13+C24C24U11+U21RD13](In0),V3=E11FE33E11E22MFE33E11SE22FNE44E33E11SU31RNE44E33+LE11U32+U33RE33,^V3=MFE44+SSE22FN+LMLSU41+LMU31RNU42RE44, $

    $ 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

    $ ^A1=C11LA11,P1=E11C11A11B11D11,^A2=C22LA22,P2=E22C22A22B22D22,A5=F11LA1L^A1,A6=F11LA11,A7=LA22L^A2,A8=LA22,B5=RD11,B7=RD22F22,B=G11F11A11B11F11LA1^A1P1D11A22B22F22LA22^A2P2D22F22,RA5A6=A11,RA5A7=A2,RA5A8=A33,F22LB5=N1,RA11A2=M1,S1=A2LM1,RA5B=T1,C=RM1RA11,^C1=CA33,^C2=RA11A33,^C11=RA2A33,^C4=A33,D=LB5LN1,^D1=B7,^D2=B7LF22,^D3=B7LB5,^D4=B7D,^E1=CT1,^E2=RA11T1LF22,^E11=RA2T1LB5,^E4=T1D,C1=(L^C2,L^C4),D13=(R^D1R^D3),D1=L^C1,D2=R^D2,C33=L^C11,D33=R^D4,E11=RC1D1,E2=RC1C33,E33=D2LD13,E44=D33LD13,M=RE11E2,N=E44LE33,F=^F2^F1,E=RC1FLD13,S=E2LM,F11=^C2L^C1,^G1=^E2^C2^C1^E1^D1^D2,F33=^C4L^C11,^G2=^E4^C4^C11^E11^D3^D4,^F1=^C1^E1^D1+L^C1^C2^E2^D2,^F2=^C11^E11^D3+L^C11^C4^E4^D4. $

    Then, the following statements are equivalent:

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

    $ \mathrm{(2)} $

    $ RA11B11=0,R^A1P1=0,P1LD11=0,RA22B22=0,R^A2P2=0,P2LD22=0, R^Ci^Ei=0,^EiL^Di=0(i=¯1,2),R^C11^E11=0,R^C4^E4=0,^E11L^D3=0,^E4L^D4=0,RE11ELE44=0. $

    $ \mathrm{(3)} $

    $ r(B11,A11)=r(A11),r(E11C11B11D11A11)=r(C11A11),r(E11D11)=r(D11), r(B22,A22)=r(A22),r(E22C22B22D22A22)=r(C22A22),r(E22D22)=r(D22),r(B11A11A22G11B22F22A22F11)=r(A11A22F11),r(F220B11A11A22G11A22F11)=r(F22)+r(A11A22F11),r(C11E11A11B11D11A22F11B22F22D11A22G11D11)=r(C11A11A22F11),r(0F22D11C11E11A11B11D11A22F11A22G11D11)=r(C11A11A22F11)+r(F22D11),r(F220D22C22G11C22F11E22B11A110A22G11A22F11B22D22)=r(F22,D22)+r(C22F11A22F11A11),r(G11F11F220B11A11)=r(F11A11)+r(F22),r(0D22F22D11C22F11E22C22G11D11C110E22A22F110A22G11D11B22F22D11A110B11D11)=r(C22F11C11A22F11A11)+r(D22,F22D11),r(F11G11D110F22D11C11E11A11B11D11)=r(F11C11A11)+r(F22D11), $
    $ r(F22000000F22D110B22B11A11000C22G11C22F11C22G11D11C22F11E2200E11C110A22G11A22F11A22G11D11B22F22D11A22F11000B11D11A110A22G11A22F11000)=r(F22000F22D11D22)+r(C22F11C22F11A22F11A22F110C110A11A110A110A22F110). $

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

    $ X1=A11B11+LA1^A1P1^B1+LA1L^A1W1+LA1W2R^B1,X2=A2B22+LA2^A2P2^B2+LA2L^A2W3+LA3W4R^B2, $

    where

    $ W1=A5(BA6W1B5A7W3F22A8W4B7)+LA5U1,W2=A1TB5A1A2M1TB5A1S1A2TN1F22B5A1S1U2RN1F22B5+LA1U3+U4RB5,W3=M1TF22+S1S1A2TN1+LM1LS1U5+U6RF22+LM1U2RN1,W4=^F1+L^C2V1+V2R^D1+L^C1V3R^D2, or W4=^F2L^C4^V1^V2R^D3L^C11^V3R^D4, $

    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} $,

    $ V1=(Im,0)[C1(FD1V3D2C33^V3D33)C1U11D1+LC1U12],^V1=(0,Im)[C1(FD1V3D2C33^V3D33)C1U11D1+LC1U12],^V2=[RC1(FD1V3D2C33^V3D33)D1+C1C1U11+U21RD1](0In),V2=[RC1(FC2V3D2C33^V3D33)D1+C1C1U11+U21RD1](In0),V3=E11FE33E11E2MFE33E11SE2FNE44E33E11SU31RNE44E33+LE11U32+U33RE33,^V3=MFE44+SSE2FN+LMLSU41+LMU31RNU42RE44, $

    $ 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.

    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

    $ A1=C11LA11,P1=E11C11A11B11Cη11,B2=Aη1,P2=Pη1,ˆB1=RB2(F22LA11)η,ˆA3=F11LA11,ˆA2=ˆA3LA1,ˆA4=LC11,ˆB3=(F11ˆA4)η,ˆB4=(F22LA11)η,Fη11LˆB1=ˆB11,P=G11F11A11B11Fη11ˆA3A1P1(F11C11)η(F22A11B11)ηC11P2B2ˆB4,ˆB22LB11=N1,ˆB3LˆB1=ˆB22,ˆB4LˆB1=ˆB33,RˆA2ˆA3=ˆM1,S1=ˆA3LM1,T1=PL^B1,C=RM1RˆA2,C1=CˆA4,C2=RˆA2ˆA4,C3=RˆA3ˆA4,C4=ˆA4,D=LˆB11LN1,D1=ˆB33,D2=ˆB33LˆB22,D4=ˆB33D,E1=CT1,E2=RˆA2T1LˆB11,E4=T1D,ˆC11=(LC2,LC4),D3=ˆB33LˆB11,ˆD11=(RD1RD3),ˆC22=LC1,ˆD22=RD2, ˆC33=LC3,ˆD33=RD4,ˆE11=RˆC11ˆC22,ˆE22=RˆC11ˆC33,ˆE33=ˆD22LˆD11,ˆE44=ˆD33LˆD11,M=RˆE11ˆE22,N=ˆE44LˆE33, F=F2F1,E=RˆC11FLˆD11,S=ˆE22LM,^F11=C2LC1,G1=E2C2C1E1D1D2,^F22=C4LC3,G2=E4C4C3E3D3D4,F1=C1E1D1+LC1C2E2D2,F2=C3E3D3+LC3C4E4D4. $

    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)} $

    $ r(B11,A11)=r(A11),r(E11C11B11Cη11A11)=r(C11A11), r(E11Cη11)=r(C11),r(Fη220Cη11Aηη11B11Fη11A1100C11G11C11F11Eη11C11Bη11)=r(Fη22,Cη11,Aη11)+r(A11C11F11),r(Fη110Cη1100Fη2200Cη11Aη110C11E11000A11B11Cη1100C11G11C11F110Eη11C11Bη11)=r(C11A110)+r(Fη11Cη1100Fη220Cη11Aη11),r(Fη11000Fη220Cη11Aη110A1100C11G11C11F11Eη11C11Bη11)=r(Fη1100Fη22Cη11Aη11)+r(A11C11F11),r(Fη1100Fη22Cη11Aη11C11G11Eη11C11Bη11)=r(Fη1100Fη22Cη11Aη11,),r(G11F11Bη11Fη220Aη11B11Fη11A110)=r(F11A11)+r(Fη22,Aη11),r(G11F110Bη11Fη110Cη110Fη2200Aη110C11E1100A11B11Cη110)=r(Fη11Cη110Fη220Aη11)+r(F11C11A11),r(G11F11Bη11Fη1100Fη220Aη110A110)=r(Fη110Fη22Aη11)+r(F11A11),r(G11Bη11Fη110Fη22Aη11)=r(Fη110Fη22Aη11),r(Fη11000000Cη110Fη220000Cη11Aη110000Fη1100000000Fη22Cη11Aη110000Fη220Fη2200000Aη110C1100000E1100A1100000B11Cη110C11G11C11F11000Eη11C11Bη1100)=r(Fη1100000Cη110Fη22000Cη11Aη11000Fη110000000Fη22Cη11Aη110000Fη22Fη2200000Aη11)+r(C11A11C11F11). $

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

    $ A11^X1=B11,C11^X1Cη11=E11,^X2Aη11=Bη11,C11^X2Cη11=Eη11,F11X1Fη11+^X2ηFη22=G11. $ (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

    $ X1:=^X1+Xη22. $

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

    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.

    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.

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

    The authors declare that they have no conflicts of interest.

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