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Review Topical Sections

Properties, performance, and applications of biofuel blends: a review

  • Received: 15 May 2017 Accepted: 28 July 2017 Published: 03 August 2017
  • Biofuels such as ethanol and biodiesel derived from living plants or animal matter can be used directly in their neat forms or as blends with their fossil counterparts in internal combustion engines. Although the properties and performance of neat biofuels have been extensively reported, this is not the case for many blends. The purpose of this review is to analyze different forms of biofuel blends that are under research and development comparing their utility and performance in the two primary classes of engines, i.e., spark ignition and compression ignition engines. The fuel properties, performance and emission characteristics, advantages and disadvantages of various fuel blends are compared and discussed. The analysis reveals certain blends possess better overall fuel properties and yield better overall performance than the neat or fossil forms.

    Citation: Husam Al-Mashhadani, Sandun Fernando. Properties, performance, and applications of biofuel blends: a review[J]. AIMS Energy, 2017, 5(4): 735-767. doi: 10.3934/energy.2017.4.735

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  • Biofuels such as ethanol and biodiesel derived from living plants or animal matter can be used directly in their neat forms or as blends with their fossil counterparts in internal combustion engines. Although the properties and performance of neat biofuels have been extensively reported, this is not the case for many blends. The purpose of this review is to analyze different forms of biofuel blends that are under research and development comparing their utility and performance in the two primary classes of engines, i.e., spark ignition and compression ignition engines. The fuel properties, performance and emission characteristics, advantages and disadvantages of various fuel blends are compared and discussed. The analysis reveals certain blends possess better overall fuel properties and yield better overall performance than the neat or fossil forms.


    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 Aii, Bii, Cii, Dii, Eii, Fii(i=¯1,2), H11, and G11 are known matrices, while Xi(i=¯1,2) are unknown.

    In this paper, R and Hm×n denote the real number field and the set of all quaternion matrices of order m×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. 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 (MP) inverse. It should be noted that A is used to represent the MP inverse of A. Additionally, LA=IAA and RA=IAA denote two projectors toward A.

    H is known to be an associative noncommutative division algebra over 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 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 MP 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 MP 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 MP 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 η-Hermitian matrices has gained significant attention [17]. An η-Hermitian matrix, for η{i,j,k}, is defined as a matrix A such that A=Aη, where Aη=ηAη. 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, η-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 MP 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 MP inverse and the rank equality of matrices to investigate the necessary and sufficient conditions for the solvability of quaternion matrix equations involving η-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 MP 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 H.

    Lemma 2.1. [18] Let AHm×n, BHm×k, CHl×n, DHj×k, and EHl×i be given. Then, the following rank equality holds:

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

    Lemma 2.2. [19] Let AHm×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 A1 and A2 be given quaternion matrices with adequate shapes. Then, the equation A1X=A2 is solvable if, and only if, A2=A1A1A2. In this case, the general solution to this equation can be expressed as

    X=A1A2+LA1U1,

    where U1 is any matrix with appropriate size.

    Lemma 2.4. [20] Let A1 and A2 be given quaternion matrices with adequate shapes. Then, the equation XA1=A2 is solvable if, and only if, A2=A2A1A1. In this case, the general solution to this equation can be expressed as

    X=A2A1+U1RA1,

    where U1 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 Ci,Di, and Z(i=¯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:

    (1) Equation (2.1) is consistent.

    (2)

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

    (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=T1C4Y3D4,Ui(i=¯1,8) are arbitrary matrices with appropriate sizes over 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,

    U11,U12,U21,U31,U32,U33,U41, and U42 are arbitrary quaternion matrices with appropriate sizes, and m and n denote the column number of C4 and the row number of D4, 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 Aii,Bii,Cii,Dii,Eii,Fii,H11, and G11 (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:

    (1) System (1.1) is solvable.

    (2)

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

    (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)(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 A1,P1,B2,P2 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 ^Bi,^Aj(i=¯1,4,j=¯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 Ci,Ei,Di(i=¯1,4),^E11,^E44 are defined as (3.2), T=T1^A4W3^B4,Vi(i=¯1,8) are arbitrary matrices with appropriate sizes over 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,

    U11,U12,U21,U31,U32,U33,U41, and U42 are any quaternion matrices with appropriate sizes, and m and n denote the column number of C22 and the row number of A22, 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)(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 X01,X02,U01, and U02 such that

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

    holds, where

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

    P1=E11C11X01D11,P2=E22C22X02D22, and P=G11F11X01H11F11LA11U01H11X02F22U02RA22F22.

    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)(3.7),(3.28)(3.8),(3.29)(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)(3.11),(3.32)(3.12),(3.33)(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=T1^A4W3^B4,Vi(i=¯4,8) are arbitrary matrices with appropriate sizes over H, U11,U12,U21, U31,U32,U33,U41, and U42 are any quaternion matrices with appropriate sizes, and m and n denote the column number of C22 and the row number of A22, respectively.

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

    Corollary 3.3. [16] Let Aii,Bii,Cii,Dii,Eii,Fii(i=1,2), and G11 be given matrices with appropriate dimensions over 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:

    (1) Equation (1.8) is consistent.

    (2)

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

    (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 H, where Aii,Bii,Cii,Dii,Eii,Fii,G11, and H11(i=¯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:

    (1) System (1.2) is consistent.

    (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.

    (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=T1H11W4^B4,Ui(i=¯1,6) are arbitrary matrices with appropriate sizes over 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,

    U11,U12,U21,U31,U32,U33,U41, and U42 are any quaternion matrices with appropriate sizes, and m and n denote the column number of H11 and the row number of A22, respectively.

    Theorem 4.2. Consider the matrix equation (1.3) over H, where Aii,Bii,Cii,Dii,Eii,Fii,G11H11(i=¯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:

    (1) System (1.3) is consistent.

    (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.

    (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=T1^A22W4^B4,Ui(i=¯1,6) are arbitrary matrices with appropriate sizes over 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,

    U11,U12,U21,U31,U32,U33,U41, and U42 are any matrices with appropriate sizes, and m and n denote the column number of H11 and the row number of D22, respectively.

    Theorem 4.3. Consider the matrix equation (1.4) over H, where Aii,Bii,Cii,Dii,Eii,Fii(i=¯1,2), and G11 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:

    (1) Equation (1.4) is consistent.

    (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.

    (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=T1A8W4B7,Ui(i=¯1,6) are arbitrary matrices with appropriate sizes over 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,

    U11,U12,U21,U31,U32,U33,U41, and U42 are any quaternion matrices with appropriate sizes, and m and n denote the column number of A22 and the row number of D22, respectively.

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

    Theorem 5.1. Let A11,B11,C11,E11,F11,F22, and G11(G11=Gη11) 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:

    (1) System (1.9) is solvable.

    (2)

    RA11B11=0,RA1P1=0,P1(RC11)η=0,RCiEi=0,EiLDi=0(i=¯1,4),RˆE11ELˆE44=0.

    (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, X1, then (^X1, ^X2):=(X1, Xη1) is a solution of (5.1). Conversely, if (5.1) has a solution, say (^X1, ^X2), 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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