AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi 10.1155/2019/5926832 5926832 Research Article Determinantal Representations of General and (Skew-)Hermitian Solutions to the Generalized Sylvester-Type Quaternion Matrix Equation http://orcid.org/0000-0001-8426-0026 Kyrchei Ivan I. 1 Wong Patricia J. Y. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics NAS of Ukraine Lviv 79060 Ukraine nas.gov.ua 2019 612019 2019 30 06 2018 18 12 2018 612019 2019 Copyright © 2019 Ivan I. Kyrchei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we derive explicit determinantal representation formulas of general, Hermitian, and skew-Hermitian solutions to the generalized Sylvester matrix equation involving -Hermicity AXA+BYB=C over the quaternion skew field within the framework of the theory of noncommutative column-row determinants.

1. Introduction

Let Hm×n and Hrm×n stand for the set of all m×n matrices and for its subset of matrices with rank r, respectively, over the quaternion skew field(1)H=a0+a1i+a2j+a3ki2=j2=k2=-1,ij=-ji=k,a0,a1,a2,a3R,where R is the real number field. For AHm×n, the symbol A stands for conjugate transpose (Hermitian adjoint) of A. A matrix AHn×n is Hermitian if A=A.

The Moore-Penrose inverse of AHm×n is called the unique matrix XHn×m satisfying the following four equations(2)1.AXA=A,2.XAX=X,3.AX=AX,4.XA=XA.It is denoted by A.

The two-sided generalized Sylvester matrix equation(3)AXB+CYD=E has been well studied in matrix theory. For instance, Huang  obtained necessary and sufficient conditions for the existence of solutions to (3) with X=Y over the quaternion skew field. Baksalary and Kala  derived the general solution to (3) expressed in terms of generalized inverses which has been extended to an arbitrary division ring and on any regular ring with identity in [3, 4]. Ranks and independence of solutions to (3) were explored in . In  expressions, as well as necessary and sufficient conditions, were given for the existence of the real and pure imaginary solutions to the consistent quaternion matrix equation (3).

The high research activities on Sylvester-type matrix equations can be observed lately. In particular, we note the following papers concerning methods of their computing solutions. Liao et al.  established a direct method for computing its approximate solution using the generalized singular value decomposition and the canonical correlation decomposition. Efficient iterative algorithms were presented to solve a system of two generalized Sylvester matrix equations in  and to solve the minimum Frobenius norm residual problem for a system of Sylvester-type matrix equations over generalized reflexive matrix in .

Systems of periodic discrete-time coupled Sylvester quaternion matrix equations , systems of quaternary coupled Sylvester-type real quaternion matrix equations , and optimal pole assignment of linear systems by the Sylvester matrix equations  have been explored. Some constraint generalized Sylvester matrix equations [13, 14] were studied recently.

Special solutions to Sylvester-type quaternion matrix equations have been actively studied. Roth’s solvability criteria for some Sylvester-type matrix equations were extended over the quaternion skew field with a fixed involutive automorphism in . Şimşek et al.  established the precise solutions on the minimum residual and matrix nearness problems of the quaternion matrix equation (AXB,DXE)=(C,F) for centrohermitian and skew-centrohermitian matrices. Explicit solutions to some Sylvester-type quaternion matrix equations (with j-conjugation) were established by means of Kronecker map and complex representation of a quaternion matrix in [17, 18]. The expressions of the least squares solutions to some Sylvester-type matrix equations over nonsplit quaternion algebra  and Hermitian solutions over a split quaternion algebra  were derived. Solvability conditions and general solution for some generalized Sylvester real quaternion matrix equations involving η-Hermicity were given in [21, 22].

Many authors have paid attention also to the Sylvester-type matrix equation involving -Hermicity(4)AXA+BYB=C. Chang and Wang  derived expressions for the general symmetric solution and the general minimum-2-norm symmetric solution to the matrix equation (4) within the real settings. Xu et al.  have given a representation of the least-squares Hermitian (skew-Hermitian) solution to the matrix equation (4). Zhang  obtained a representation of the general Hermitian nonnegative-definite (respectively positive-definite) solution to (4) within the complex settings. Yuan et al.  derived the expression of Hermitian solution for the matrix nearness problem associated with the quaternion matrix equation (4). Wang et al.  gave a necessary and sufficient condition for the existence and an expression for the re-nonnegative definite solution to (4) over H by using the decomposition of pairwise matrices. Wang et al.  established the extreme ranks for the general (skew-)Hermitian solution to (4) over H.

Motivated by the vast application of quaternion matrices and the latest interest of Sylvester-type quaternion matrix equations, the main goal of the paper is to derive explicit determinantal representation formulas of the general, Hermitian, and skew-Hermitian solutions to (4) based on determinantal representations of the Moore-Penrose inverse.

Determinantal representation of a solution gives a direct method of its finding analogous to classical Cramer’s rule that has important theoretical and practical significance. However, determinantal representations are not so unambiguous even for generalized inverses within the complex or real settings. Through looking for their more applicable explicit expressions, there are various determinantal representations of generalized inverses (see, e.g., ). By virtue of noncommutativity of quaternions, the problem for determinantal representation of generalized quaternion inverses is even more complicated, and only now it can be solved due to the theory of column-row determinants introduced in [32, 33]. Within the framework of the theory of row-column determinants, determinantal representations of various generalized quaternion inverses, namely, the Moore-Penrose inverse , the Drazin inverse , the W-weighted Drazin inverse , and the weighted Moore-Penrose inverse , have been derived by the author. These determinantal representations were used to obtain explicit representation formulas for the minimum norm least squares solutions  and weighted Moore-Penrose inverse solutions  to some quaternion matrix equations and explicit determinantal representation formulas of both Drazin and W-weighted Drazin inverse solutions to some restricted quaternion matrix equations and quaternion differential matrix equations . Recently, determinantal representations of solutions to some systems of quaternion matrix equations [43, 44] and, in , two-sided generalized Sylvester matrix equation (3) have been derived by the author as well.

Other researchers also used the row-column determinants in their developments. In particular, Song derived determinantal representations of the generalized inverse AT,S2  and the Bott-Duffin inverse . Song et al. obtained the Cramer rules for the solutions of restricted matrix equations  and for the generalized Stein quaternion matrix equation , and so forth. Moreover, Song et al.  have just recently considered determinantal representations of the general solution to the generalized Sylvester matrix equation (3) over H using row-column determinants as well. But their approach differs from ours because for determinantal representations of solutions we use only coefficient matrices of the equation, while in  supplementary matrices have been constructed and used.

The paper is organized as follows. In Section 2, we start with some remarkable results which have significant role during the construction of the main results of this paper. Elements of the theory of row-column determinants are given in Section 2.1, determinantal representations of the Moore-Penrose inverse and of the general solution to the quaternion matrix equation AXB=C and its special cases are considered in Section 2.2, and the explicit determinantal representation of the general solution to (3) previously obtained within the framework of the theory of row-column determinants is in Section 2.3. The main results of the paper, namely, explicit determinantal representation formulas of the general, Hermitian, skew-Hermitian solutions to (4), are derived in Section 3. In Section 4, a numerical example to illustrate the main results is considered. Finally, in Section 5, the conclusions are drawn.

2. Preliminaries

We commence with the following preliminaries which have crucial function in the construction of the chief outcomes of the following sections.

2.1. Elements of the Theory of Row-Column Determinants

Due to noncommutativity of quaternions, a problem of defining a determinant of matrices with noncommutative entries (which is also defined as noncommutative determinants) has been unsolved for a long time. There are several versions of the definition of noncommutative determinant (see, e.g., ). But any of the previous noncommutative determinants has not fully retained those properties which it has owned for matrices with commutative entries. Moreover, if functional properties of noncommutative determinant over a ring are satisfied, then it takes on a value in its commutative subset. This dilemma can be avoided thanks to the theory of row-column determinants.

Suppose Sn is the symmetric group on the set In={1,,n}. Let AHn×n. Row determinants of A along each row can be defined as follows.

Definition 1 (see [<xref ref-type="bibr" rid="B32">32</xref>]).

The i th row determinant of A=(aij)Hn×n is defined for all i=1,,n by putting(5)rdetiA=σSn-1n-raiik1aik1ik1+1aik1+l1iaikrikr+1aikr+lrikr,σ=iik1ik1+1ik1+l1ik2ik2+1ik2+l2ikrikr+1ikr+lr,where ik2<ik3<<ikr and ikt<ikt+s for all t=2,,r and s=1,,lt.

Similarly, for a column determinant along an arbitrary column, we have the following definition.

Definition 2 (see [<xref ref-type="bibr" rid="B32">32</xref>]).

The j th column determinant of A=(aij)Hn×n is defined for all j=1,,n by putting(6)cdetjA=τSn-1n-rajkrjkr+lrajkr+1jkrajjk1+l1ajk1+1jk1ajk1j,τ=jkr+lrjkr+1jkrjk2+l2jk2+1jk2jk1+l1jk1+1jk1j,where jk2<jk3<<jkr and jkt<jkt+s for t=2,,r and s=1,,lt.

So an arbitrary n×n quaternion matrix inducts a set from n row determinants and n column determinants that are different in general. Only for Hermitian A, we have ,(7)rdet1A==rdetnA=cdet1A==cdetnAR,which enables defining the determinant of a Hermitian matrix by putting(8)detArdetiA=cdetiAfor all i=1,,n.

Its properties are similar to the properties of an usual (commutative) determinant and they have been completely explored in  by using row and column determinants that are so defined only by construction. We note the following that will be required below.

Lemma 3.

Let AHm×n. Then cdetiA=rdetiA¯, rdetiA=cdetiA¯.

2.2. Determinantal Representations of the Moore-Penrose Inverse with Applications to Some Quaternion Matrix Equations

For introducing determinantal representations of the Moore-Penrose inverse, the following notations will be used.

Let αα1,,αk1,,m and ββ1,,βk1,,n be subsets of the order 1kminm,n. Aβα denotes a submatrix of A whose rows are indexed by α and the columns indexed by β. So Aαα denotes a principal submatrix of A with rows and columns indexed by α. If AHn×n is Hermitian, then Aαα denotes the corresponding principal minor of detA.

Let Lk,nα:α=α1,,αk,1α1<<αkn denote the collection of strictly increasing sequences of k integers chosen from 1,,n for all 1kn. Then, for fixed iα and jβ, the collection of sequences of row indexes that contain the index i is denoted by Ir,miα:αLr,m,iα; similarly, the collection of sequences of column indexes that contain the index j is denote by Jr,njβ:βLr,n,jβ.

Let a.j be the jth column and ai. be the ith row of A. Suppose A.jb denote the matrix obtained from A by replacing its jth column with the column b and Ai.b denote the matrix obtained from A by replacing its ith row with the row b. Denote by a.j and ai. the jth column and the ith row of A, respectively.

Theorem 4 (see [<xref ref-type="bibr" rid="B34">34</xref>]).

If AHrm×n, then the Moore-Penrose inverse A=aijHn×m have the following determinantal representations,(9)aij=βJr,nicdetiAA.ia.jβββJr,nAAββ,(10)aij=αIr,mjrdetjAAj.ai.αααIr,mAAαα.

Remark 5.

For an arbitrary full-rank matrix AHrm×n, a column vector bHn×1 and a row vector cH1×m we put(11)cdetiAA.ib=βJn,nicdetiAA.ibββ,detAA=βJn,nAAββwhenr=n,rdetjAAj.c=αIm,mjrdetjAAj.cαα,detAA=αIm,mAAααwhenr=m.

Remark 6.

First note that (A)=(A). Because of symbol equivalence, we shall use the denotation A,(A) as well. So by Lemma 3, for the Hermitian adjoint matrix AHrn×m determinantal representations of its Moore-Penrose inverse (A)=(aij)Hm×n are(12)aij=aji¯=αIr,njrdetjAAj.ai.αααIr,nAAαα,(13)aij=βJr,micdetiAA.ia.jβββJr,mAAββ.

Since the projection matrices AAPA=pij and AAQA=qij are Hermitian, then pij=pji¯ and qij=qji¯ for all ij. So due to Theorem 4 and Remark 6 we have evidently the following corollaries.

Corollary 7.

If AHrm×n, then the projection matrix PA=pijn×n has the determinantal representations(14)pij=βJr,nicdetiAA.ia˙.jβββJr,nAAββ=αIr,njrdetjAAj.a˙i.αααIr,nAAαα, where a˙.j and a˙i. are the jth column and ith row of AAHn×n, respectively.

Corollary 8.

If AHrm×n, then the projection matrix AAQA=qijm×m has the determinantal representation(15)qij=αIr,mjrdetjAAj.a¨i.αααIr,mAAαα=βJr,micdetiAA.ia¨.jββαJr,mAAββ, where a¨i. and a¨j. are the ith row and the jth column of AAHm×m.

Determinantal representations of orthogonal projectors LAI-AA and RAI-AA induced from A can be derived similarly.

Theorem 9 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Let AHm×n, BHr×s, CHm×s be known and XHn×r be unknown. Then the matrix equation(16)AXB=Cis consistent if and only if AACBB=C. In this case, its general solution can be expressed as(17)X=ACB+LAV+WRB,where V,W are arbitrary matrices over H with allowable dimensions.

Theorem 10 (see [<xref ref-type="bibr" rid="B35">35</xref>]).

Let AHr1m×n, BHr2r×s. Then the partial solution X0=ACB=(xij0)Hn×r to (16) has determinantal representations,(18)xij0=βJr1,nicdetiAA.id.jBβββJr1,nAAββαIr2,rBBαα,or(19)xij0=αIr2,rjrdetjBBj.di.AααβJr1,nAAββαIr2,rBBαα,where(20)d.jB=αIr2,rjrdetjBBj.c~k.ααHn×1,k=1,,n,di.A=βJr1,nicdetiAA.ic~.lββH1×r,l=1,,r,are the column vector and the row vector, respectively. c~i. and c~.j are the ith row and the jth column of  C~=ACB.

Corollary 11.

Let AHkm×n, CHm×s be known and XHn×s be unknown. Then the matrix equation AX=C is consistent if and only if AAC=C. In this case, its general solution can be expressed as X=AC+LAV, where V is an arbitrary matrix over H with an allowable dimension. The partial solution X0=AC has the following determinantal representation,(21)xij0=βJk,nicdetiAA.ic^.jβββJk,nAAββ.where c^.j is the jth column of  C^=AC.

Corollary 12.

Let BHkr×s, CHn×s be given and XHn×r be unknown. Then the equation XB=C is solvable if and only if  C=CBB and its general solution is X=CB+WRB, where W is any matrix with an allowable dimension. Moreover, its partial solution X=CB has the determinantal representation, (22)xij=αIk,rjrdetjBBj.c^i.αααIk,rBBαα. where c^i. is the ith row of  C^=CB.

2.3. Determinantal Representations of the General Solution to the Sylvester Matrix Equation (<xref ref-type="disp-formula" rid="EEq1.1">3</xref>) Lemma 13 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Let AHm×n, BHr×s, CHm×p, DHq×s, EHm×s. Put M=RAC, N=DLB, S=CLM. Then the following results are equivalent.

Eq. (3) has a pair solution (X,Y), where XHn×r, YHp×q.

RMRAE=0, RAELD=0, ELDLN=0, RCELB=0.

QMRAEPD=RAE, QCELBPN=ELB.

rankACE=rankAC, rankBDE=rankBD, rankAE0D=rankA00D, rankCE0B=rankC00B.

In that case, the general solution to (3) can be expressed as(23)X=AEB-ACMRAEB-ASCELBNDB--ASVRNDB+LAU+ZRB,(24)Y=MRAED+LMSSCELBN+LMV-SSVNN+WRD,where U, V, Z, and W are arbitrary matrices over H obeying agreeable dimensions.

Some simplifications of (23) and (24) can be derived due to the quaternionic analog of the following proposition.

Lemma 14 (see [<xref ref-type="bibr" rid="B57">57</xref>]).

If AHn×n is Hermitian and idempotent, then for any matrix BHm×n the following equations hold(25)ABA=BA,ABA=AB.

Since RA, LB, and LM are projectors, then by Lemma 14 the simplifications of (23) and (24) are as follows: (26)X=AEB-ACMEB-ASCENDB-ASVRNDB++LAU+ZRB,Y=MED+PSCEN+LMV-PSVQN+WRD. By putting U,V,Z, and W as zero-matrices, we obtain the partial solution to (3),(27)X=AEB-ACMEB-ASCENDB,(28)Y=MED+PSCEN.The following theorem gives determinantal representations of (27)-(28).

Theorem 15 (see [<xref ref-type="bibr" rid="B45">45</xref>]).

Let AHr1m×n, BHr2r×s, CHr3m×p, DHr4q×s, rankM=r5, rankN=r6, rankS=r7. Then the pair solution (27)-(28), X=xijHn×r, Y=xgfHp×q, to (3) by the components (29)xij=xij1-xij2-xij3,ygf=ygf1+ygf2,has the determinantal representation, as follows.

(i)(30)xij1=βJr1,nicdetiAA.id.jBβββJr1,nAAββαIr2,rBBαα,or (31)xij1=αIr2,rjrdetjBBj.di.AααβJr1,nAAββαIr2,rBBαα, where (32)d.jB=αIr2,rjrdetjBBj.ek.1ααHn×1,k=1,,n,di.A=βJr1,nicdetiAA.ie.l1ββH1×r,l=1,,r, are the column vector and the row vector, respectively. ek.(1) and e.l(1) are the kth row and the lth column of E1AEB.

(ii) (33)xij2=t=1qφiqαIr2,rjrdetjBBj.eq.2ααβJr1,nAAβββIr5,mMMαααIr2,rBBαα, where eq.(2) is qth row of E2EB. (34)φiq=βJr1,nicdetiAA.iψ.qMββ=αIr5,mqrdetqMMq.ψi.Aαα,ψ.qM=αIr5,mqrdetqMMq.cf.1ααHn×1,f=1,,n,ψi.A=βJr1,nicdetiAA.ic.s1ββH1×m,s=1,,m,are the column vector and the row vector, respectively. cf.(1) and c.s(1) are the fth row and the sth column of C1ACM.

(iii)(35)xij3==t=1pf=1qβJr1,nicdetiAA.is.t1βββJr3,ptcdettCC.te.f3ββηfjβJr1,nAAβββJr3,pCCββαJr6,sNNββαIr2,rBBαα,where s.t(1) is the tth column of S1AS, e.f(3) is the fth column of E3CE, and (36)ηfj=αIr2,rtrdetjBBj.ζf.Nαα=βJr6,sfcdetfNN.fζ.jBββ,ζf.N=βJr6,sfcdetfNN.fd.k1ββH1×r,k=1,,r,ζ.jB=αIr2,rjrdetjBBj.dl.1ααHs×1,l=1,,s, are the row vector and the column vector, respectively. d.k(1) and dl.(1) are the kth column and the lth row of D1=NDB.

(iv) (37)ygf1=βJr5,pgcdetgMM.gd.fDβββJr5,pMMββαIr4,qDDαα,

or(38)ygf1=αIr4,qfrdetfDDf.dg.MααβJr5,pMMββαIr4,qDDαα,

where (39)d.fD=αIr4,qfrdetfDDf.ek.4ααHp×1,k=1,,p,dg.M=βJr5,pgcdetgMM.ge.l4ββH1×q,l=1,,q, are the column vector and the row vector, respectively. ek.(4) and e.l(4) are the kth row and the lth column of E4MED.

(v) (40)ygf2=t=1pβJr7,pgcdetgSS.gs¨.tββξtfβJr7,pSSβββJr3,pCCββαIr6,qNNαα,where (41)ξtf=αIr6,qtrdetfNNf.ϕt.Cαα=βJr3,ptcdettCC.tϕ.fNββ,ϕt.C=βJr3,ptcdettCC.te.k5ββH1×q,k=1,,q,ϕ.fN=αIr6,qjrdetfNNf.el.5ααHs×1,l=1,,p, are the row vector and the column vector, respectively. e.k(5) and el.(5) are the kth column and the lth row of E5=CEN.

3. Determinantal Representations of the General and (Skew-)Hermitian Solutions to (<xref ref-type="disp-formula" rid="EEq1.2">4</xref>)

Now consider (4). Since for an arbitrary matrix A it is evident that PA=AA=AA=QA, so QA=PA, LA=I-PA=I-QA=RA, and RA=LA. Due to the above, M=RAB and N=BLA=BRA=(RAB)=M, and we obtain the following analog of Lemma 13.

Lemma 16.

Let AHm×n, BHm×k, CHm×m. Put M=RAB, S=BLM. Then the following results are equivalent.

Equation (4) has a pair solution (X,Y), where XHn×n, YHk×k.

RMRAC=0, RACRB=0, CRBRM=0, RBCRA=0.

QMRACQB=RAC, QBCRAQM=CRA.

rankABC=rankAB, rankAC0B=rankA00B, rankBC0A=rankB00A.

In that case, the general solution to (4) can be expressed as follows: (42)X=ACA,-ABMCA,-ASB,CM,BA,-ASVLMBA,+LAU+ZLA,Y=MCB,+PSBCM,+LMV-PSVPM+WLB. where U, V, Z, and W are arbitrary matrices over H with allowable dimensions.

By putting U, V, Z, and W as zero-matrices with compatible dimensions, we obtain the following partial solution to (4),(43)X=ACA,-ABMCA,-ASBCM,BA,,(44)Y=MCB,+PSBCM,.The following theorem gives determinantal representations of (43)-(44).

Theorem 17.

Let AHr1m×n, BHr2m×k, rankM=r3, rankS=r4. Then the partial pair solution (43)-(44) to (4), X=xijHn×n, Y=ypgHk×k, by the components(45)xij=xij1-xij2-xij3,ypg=ypg1+ypg2,possesses the following determinantal representations:

(i)(46)xij1=αIr1,njrdetjAAj.vi.αααIr1,nAAαα2or(47)xij1=βJr1,nicdetiAA.iv.jβββJr1,nAAββ2,where(48)vi.=βJr1,nicdetiAA.ic.s1ββH1×n,s=1,,n,(49)v.j=αIr1,njrdetjAAj.cf.1ααHn×1,f=1,,n are the row vector and the column vector, respectively; c.s(1) and cf.(1) are the sth column and the fth row of C1=ACA.

(ii)(50)xij2=αIr1,njrdetjAAj.ϕ~i.ααβJr1,nAAββ2αIr3,mMMαα,where ϕ~i. is the ith row of Φ~ΦCA and Φ=(ϕiq)Hn×m is such that(51)ϕiq=βJr1,nicdetiAA.iη.qMββ=αIr3,mqrdetqMMq.ηi.Aαα,(52)η.qM=αIr3,mqrdetqMMq.bf.1ααHn×1,f=1,,n,ηi.A=βJr1,nicdetiAA.ib.s1ββH1×m,s=1,,m, are the column vector and the row vector, respectively. bf.(1) and b.s(1) are the fth row and the sth column of B1=ABM and cq.(2) is the qth row of C2=CA.

(iii)(53)xij3=βJr1,nicdetiAA.iυ~.jβββJr1,nAAββ2βJr2,kBBβββJr3,mMMββ,where υ~.j is the jth column of Υ~=ASΥ, the matrix Υ=(υpj)Hk×n such that (54)υpj=βJr2,kpcdetpBB.pc~.jββ, where c~.j is the jth column of C~=BCΦ and Φ is Hermitian adjoint to Φ=(ϕiq) from (51).

(iv)(55)ypg1=βJr3,kpcdetpMM.pd.gBβββJr3,kMMββαIr2,kBBαα,or(56)ypg1=αIr2,kgrdetgBBg.dp.MααβJr3,kMMββαIr2,kBBαα,where (57)d.gB=αIr2,kgrdetgBBg.cq.4ααHk×1,q=1,,k,dp.M=βJr3,kpcdetpMM.pc.l4ββH1×k,l=1,,k, are the column vector and the row vector, respectively. cq.(4) and c.l(4) are the qth row and the lth column of C4MCB.

(v)(58)ypg2=βJr4,kpcdetpSS.pω~.gβββJr4,kSSβββJr2,kBBββαIr3,kMMαα,where Ω~=SSΩ and Ω=(ωtg) such that(59)ωtg=βJr2,ktcdettBB.td.gMββ=αIr3,kgrdetgMMg.dt.Bαα,d.gM=αIr3,kgrdetgMMg.cq.4,ααHk×1,q=1,,k,dt.B=βJr3,ktcdettBB.tc.l4,ββH1×k,l=1,,k,are the column vector and the row vector, respectively. cq.(4,) and c.l(4,) are the qth row and the lth column of C4MCB.

Proof.

The proof evidently follows from the proof of Theorem 15 by substitution corresponding matrices. For a better understanding more complete proof will be made in some points, and a few comments will be done in others.

(i) For the first term of (43), X1=ACA=(xij(1)), we have (60)xij1=l=1mt=1mailcltatj,. By using determinantal representations (9) and (13) of the Moore-Penrose inverses A and A, respectively, we obtain (61)xij1=l=1mt=1mβJr1,nicdetiAA.ia.lββcltαIr1,njrdetjAAj.at.αααIr1,nAAααβJr1,nAAββ. Suppose el. and e.l are the unit row vector and the unit column vector, respectively, such that all their components are 0, except the lth components, which are 1. Denote C1ACA. Since l=1mt=1maflcltats=cfs(1), then (62)xij1=f=1ns=1nβJr1,nicdetiAA.ie.fββcfs1αIr1,njrdetjAAj.es.αααIr1,nAAααβJr1,nAAββ If we denote by (63)visf=1nβJr1,nicdetiAA.ie.fββcfs1=βJr1,nicdetiAA.ic.s1ββ the sth component of a row vector vi.=vi1,,vin, then (64)s=1mvisαIr1,njrdetjAAj.es.αα=αIr1,njrdetjAAj.vi.αα. Further, it is evident that βJr1,nAAββ=αIr1,nAAαα, so the first term of (43) has the determinantal representation (46), where vi. is (48).

If we denote by (65)vfj2s=1ncfs1αIr1,njrdetjAAj.es.αα==αIr1,njrdetjAAj.cf.1ααthe fth component of a column vector v.j=v1j,,vnj, then(66)f=1nβJr1,nicdetiAA.ie.fββvfj=βJr1,nicdetiAA.iv.jββ. So another determinantal representation of the first term of (43) is (47), where v.j is (49).

(ii) For the second term ABMCA,X2=xij(2) of (43), we have (67)xij2=l=1mp=1kq=1mt=1mailblpmpqcqtatj,. Using determinantal representations (9) for the Moore-Penrose inverse A, (10) for M=(mpq), and (13) for A, respectively, we obtain (68)xij2=l=1mp=1kq=1mt=1mβJr1,nicdetiAA.ia.lββblpαIr3,mqrdetqMMq.mp.ααβJr1,nAAββαIr3,mMMαα××cqtαIr1,njrdetjAAj.at.αααIr1,nAAαα.Further, thinking as above in the point (i), we obtain(69)ϕiql=1mp=1kβJr1,nicdetiAA.ia.lββblp·αIr3,mqrdetqMMq.mp.αα=βJr1,nicdetiAA.iη.qMββ=αIr3,mqrdetqMMq.ηi.Aαα,where (70)η.qM=αIr3,mqrdetqMMq.bf.1ααHn×1,f=1,,n,ηi.A=βJr1,nicdetiAA.ib.s1ββH1×m,s=1,,m, are the column vector and the row vector, respectively. bf.(1) and b.s(1) are the fth row and the sth column of B1=ABM. Construct the matrix Φ=(ϕiq)Hn×m such that ϕiq are obtained by (69), and denote Φ~ΦCA. Since(71)q=1mt=1mϕiqcqtαIr1,njrdetjAAj.at.αα=αIr1,njrdetjAAj.ϕ~i.αα, where ϕ~i. is the ith row of Φ~, then we have (50).

(iii) For the third term ASBCM,BA,X3=xij(3) of (43), we use the determinantal representation (9) to A and B. Then by Corollary 11 and taking into account the fact that M,BA,=(ABM), we have (72)xij3==p=1kt=1mβJr1,nicdetiAA.is.p1βββJr2,kpcdetpBB.pc.t3ββϕtjβJr1,nAAββ2βJr2,kBBβββJr3,mMMββ, where s.p(1) is the pth column of S1AS, c.t(3) is the tth column of C3BC, ϕtj is the tjth entry of Φ that is Hermitian adjoint to Φ=(ϕiq) from (51). Denote C3Φ=BCΦ=C~. Then,(73)t=1mβJr2,kpcdetpBB.pc.t3ββϕtj=βJr2,kpcdetpBB.pc~.jββ.Construct the matrix Υ=(υpj)Hk×n such that (74)υpj=βJr2,kpcdetpBB.pc~.jββ. Denote S1Υ=ASΥΥ~=(υ~ij)Hn×n. Since(75)βJr1,nicdetiAA.is.p1ββυpj=βJr1,nicdetiAA.iυ~.jββ, it follows (53).

(iv) Due to Theorem 10 and similarly as above for the first term Y1=MCB,=(ypg(1)) of (44), we have the determinantal representations (55) and (56).

(v) Finally, for the second term Y2=PSBCM,=ypg(2) of (44) using (14) for a determinantal representation of PS, and due to Theorem 10 for BCM,, we obtain (76)ypg2=t=1kβJr4,kpcdetpSS.ps¨.tββωtgβJr4,kSSβββJr2,kBBββαIr3,kMMαα, where ϕtg are(77)ωtg=βJr2,ktcdettBB.td.gMββ=αIr3,kgrdetgMMg.dt.Bαα,(78)d.gM=αIr3,kgrdetgMMg.cq.4,ααHk×1,q=1,,k,dt.B=βJr3,ktcdettBB.tc.l4,ββH1×k,l=1,,k, are the column vector and the row vector, respectively. cq.(4,) and c.l(4,) are the qth row and the lth column of C4MCB. Construct the matrix Ω=(ωtg)Hk×k such that ωtg are obtained by (77), and denote Ω~SSΩ. Since(79)t=1kβJr4,kpcdetpSS.ps¨.tββωtg=βJr4,kpcdetpSS.pω~.gββ,it follows (58).

Due to , the following lemma can be generalized to H.

Lemma 18.

Suppose that matrices AHm×n and BHm×m and CHm×m are given with C=C=(-C). Then when (4) is solvable, it must have Hermitian (skew-Hermitian) solutions.

The general Hermitian solution to (4) can be expressed as X^=(1/2)X+X, Y^=(1/2)Y+Y, where (X,Y) is an arbitrary solution to (4). Since by Lemma 18 the existence of Hermitian solutions to (4) needs that C is Hermitian, then (80)X=ACA,-ACM,BA,-ABMCB,SA,,Y=BCM,+MCB,PS. It is evident that the determinantal representations of X^=(x^ij) and Y^=(y^ij) can be obtained as x^ij=(1/2)xij+xji¯ for all i,j=1,,n  and  y^pg=(1/2)ypg+ygp¯ for all p,g=1,,k, where xij and ypg are determined by Theorem 15 and (81)xji¯=xji1¯-xji2¯-xji3¯,ygp¯=ygp1¯+ygp2¯. Moreover, xji(γ)¯ for all γ=1,2,3 has the following determinantal representations.

(i) xji(1)¯=xij(1).

(ii)(82)xji2¯=βJr1,njcdetiAA.iϕ~.jβββJr1,nAAββ2βJr3,kMMββ, where ϕ~.j is the jth column of Φ~=ACΦ that is Hermitian adjoint to Φ~ from the point (i) of Theorem 17, and Φ=(ϕqj)Hn×m is such that(83)ϕqj=αIr1,njrdetjAAj.ζq.Mαα=βJr3,mqcdetqMM.qζ.jAββ,(84)ζq.M=βJr3,mqcdetqMM.qb.f1,ββH1×n,f=1,,n,ζ.jA=αIr1,njrdetjAAj.bs.1,ααH1×m,s=m,,1,are the row vector and the column vector, respectively. b.f(1,) and bs.(1) are the fth column and the sth row of B1=MBA.

(iii) (85)xji3¯=αIr1,njrdetjAAj.υ~i.αααIr1,nAAαα2βIr2,kBBααβIr3,mMMαα, where υ~i. is the ith row of Υ~=ΥSA, the matrix Υ=(υip)Hn×k such that (86)υip=αIr2,kprdetpBBp.c~i.αα, where c~i. is the ith row of C~=ΦCB, and Φ is obtained by (51).

Similarly, ygp(δ)¯ for all δ=1,2 has the following determinantal representations.

(i)(87)ygp1¯=αIr3,kgrdetgMMg.dp.BαααIr3,kMMααβJr2,kBBββ=βJr2,kpcdetpBB.pd.gMββαIr3,kMMααβJr2,kBBββ, where (88)dp.B=βJr2,kpcdetpBB.pc.q4,ββH1×k,q=1,,k,d.gM=αIr3,kgrdetgMMg.cl.4,ααHk×1,l=1,,k, are the row vector and the column vector, respectively. c.q(4,) and cl.(4,) are the qth column and the lth row of C4BCM.

(ii)(89)ygp2¯=αIr4,kprdetpSSp.ω~g.αααIr4,kSSααβJr2,mBBββαIr3,mMMαα, where Ω~=ΩSS and Ω=(ωgt) such that(90)ωgt=βJr3,kgcdetgMM.gd.tBββ=αIr2,ktrdettBBt.dg.Mαα,d.tB=αIr2,ktrdettBBt.cq.4ααH1×k,q=1,,k,dg.M=βJr3,kpcdetgMM.gc.l4ββHk×1,l=1,,k, are the row vector and the column vector, respectively. c.q(4) and cl.(4) are the qth column and the lth row of C4MCB.

Remark 19.

By Lemma 18, if C=-C and (4) is solvable, then it has skew-Hermitian solutions. The general skew-Hermitian solution to (4) can be expressed as X~=(1/2)X-X, Y~=(1/2)Y-Y, where (X,Y) is an arbitrary solution to (4). So due to the above one we can obtain corresponding determinantal representations of skew-Hermitian solution.

4. An Example

In this section, we give an example to illustrate our results. Let us consider the matrix equation(91)AXA+BYB=C.where (92)A=-j+k1+i1+ij-k,B=1-k2+i,C=ki-ik. Since detAA=det4-4k4k4=0, then rankA=1, and, evidently, rankB=1. By Theorem 4, one can find(93)A=18j-k1-i1-i-j+k,B=171+k2-i,M=180.5+j-k1+i-0.5j,and S=0. It is easy to check that (91) is consistent. First, we can find the solution to (91) by direct calculation. By (43), (94)X=ACA,-ABMCA,-ASBCM,BA,=0-172-2+4i+3j-4k-2-3i+2j+2k4-3i+4j-2k-2-4i-3j-4k-0=1722-4i-3j+4k2+3i-2j-2k-4+3i-4j+2k2+4i+3j+4k,and by (43) Y=MCB,+PSBCM,=(1/63)-4+8j+10k.

Now, we find the solution to (91) by determinantal representations. So, (95)MM=941j-j1,C2CA=2-1+ij+kj+k1-i,B1=ABM=126-2i+j+7k-1-7i+6j-2k-7-i-2j+6k2-6i-7j-k.Since (96)φ11=3-i+0.5j+3.5k,φ12=-0.5-3.5i+3j-k, and βJ1,2AAββ2=64, αI1,2MMαα=4.5, then (97)x11=-6-2i+j+7k-1+i+-1-7i+6j-2kj+k288=136-i18-j24+k18. So x11 obtained by Cramer’s rule and the matrix method (94) are equal.

Similarly, we can obtain for all xij, i,j=1,2 and y11.

5. Conclusions

Within the framework of the theory of row-column determinants, we have derived explicit determinantal representation formulas (analogs of Cramer’s rule) of the general, Hermitian, and skew-Hermitian solutions to the Sylvester-type matrix equation AXA+BYB=C over the quaternion skew field. To accomplish that goal, we have used determinantal representations of the Moore-Penrose matrix inverse, which were previously introduced by the author.

Data Availability

The data used to support the findings of this study are included within the article titled “Determinantal Representations of General Solutions to the Generalized Sylvester Quaternion Matrix Equation and Its Type”. The prior studies (and datasets) are cited at relevant places within the text.

Conflicts of Interest

The author declares that he has no conflicts of interest.