# Determinantal Representations of General and (Skew-)Hermitian Solutions to the Generalized Sylvester-Type Quaternion Matrix Equation

Authors: Ivan I. Kyrchei
Journal: Abstract and Applied Analysis
 (2019)

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## Abstract

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.


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## Body

## 1. Introduction

LetHm×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+a3k∣i2=j2=k2=-1,ij=-ji=k,a0,a1,a2,a3∈R,where R is the real number field. For A∈Hm×n, the symbol A∗ stands for conjugate transpose (Hermitian adjoint) of A. A matrix A∈Hn×n is Hermitian if A∗=A.The Moore-Penrose inverse ofA∈Hm×n is called the unique matrix X∈Hn×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 [1] obtained necessary and sufficient conditions for the existence of solutions to (3) with X=Y over the quaternion skew field. Baksalary and Kala [2] 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 [5]. In [6] 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. [7] 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 [8] and to solve the minimum Frobenius norm residual problem for a system of Sylvester-type matrix equations over generalized reflexive matrix in [9].Systems of periodic discrete-time coupled Sylvester quaternion matrix equations [10], systems of quaternary coupled Sylvester-type real quaternion matrix equations [11], and optimal pole assignment of linear systems by the Sylvester matrix equations [12] 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 [15]. Şimşek et al. [16] 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 [19] and Hermitian solutions over a split quaternion algebra [20] 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 [23] 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. [24] have given a representation of the least-squares Hermitian (skew-Hermitian) solution to the matrix equation (4). Zhang [25] obtained a representation of the general Hermitian nonnegative-definite (respectively positive-definite) solution to (4) within the complex settings. Yuan et al. [26] derived the expression of Hermitian solution for the matrix nearness problem associated with the quaternion matrix equation (4). Wang et al. [27] 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. [28] 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., [29–31]). 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 [34], the Drazin inverse [35], the W-weighted Drazin inverse [36], and the weighted Moore-Penrose inverse [37], have been derived by the author. These determinantal representations were used to obtain explicit representation formulas for the minimum norm least squares solutions [38] and weighted Moore-Penrose inverse solutions [39] 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 [40–42]. Recently, determinantal representations of solutions to some systems of quaternion matrix equations [43, 44] and, in [45], 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 inverseAT,S2 [46] and the Bott-Duffin inverse [47]. Song et al. obtained the Cramer rules for the solutions of restricted matrix equations [48] and for the generalized Stein quaternion matrix equation [49], and so forth. Moreover, Song et al. [50] 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 [50] supplementary matrices have been constructed and used.The paper is organized as follows. In Section2, 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., [51–56]). 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.SupposeSn is the symmetric group on the set In={1,…,n}. Let A∈Hn×n. Row determinants of A along each row can be defined as follows.Definition 1 (see [32]).
The i th row determinant of A=(aij)∈Hn×n is defined for all i=1,…,n by putting(5)rdetiA=∑σ∈Sn-1n-raiik1aik1ik1+1…aik1+l1i…aikrikr+1…aikr+lrikr,σ=iik1ik1+1…ik1+l1ik2ik2+1…ik2+l2…ikrikr+1…ikr+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 [32]).
The j th column determinant of A=(aij)∈Hn×n is defined for all j=1,…,n by putting(6)cdetjA=∑τ∈Sn-1n-rajkrjkr+lr…ajkr+1jkr…ajjk1+l1…ajk1+1jk1ajk1j,τ=jkr+lr…jkr+1jkr…jk2+l2…jk2+1jk2jk1+l1…jk1+1jk1j,where jk2<jk3<⋯<jkr and jkt<jkt+s for t=2,…,r and s=1,…,lt.So an arbitraryn×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 [32],(7)rdet1A=⋯=rdetnA=cdet1A=⋯=cdetnA∈R,which enables definingthe determinant of a Hermitian matrix by putting(8)detA≔rdetiA=cdetiAfor all i=1,…,n.Its properties are similar to the properties of an usual (commutative) determinant and they have been completely explored in [32] by using row and column determinants that are so defined only by construction. We note the following that will be required below.Lemma 3.
LetA∈Hm×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.Letα≔α1,…,αk⊆1,…,m and β≔β1,…,βk⊆1,…,n be subsets of the order 1≤k≤minm,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 A∈Hn×n is Hermitian, then Aαα denotes the corresponding principal minor of detA.LetLk,n≔α:α=α1,…,αk,1≤α1<⋯<αk≤n denote the collection of strictly increasing sequences of k integers chosen from 1,…,n for all 1≤k≤n. Then, for fixed i∈α and j∈β, the collection of sequences of row indexes that contain the index i is denoted by Ir,mi≔α:α∈Lr,m,i∈α; similarly, the collection of sequences of column indexes that contain the index j is denote by Jr,nj≔β:β∈Lr,n,j∈β.Leta.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 [34]).
IfA∈Hrm×n, then the Moore-Penrose inverse A†=aij†∈Hn×m have the following determinantal representations,(9)aij†=∑β∈Jr,nicdetiA∗A.ia.j∗ββ∑β∈Jr,nA∗Aββ,(10)aij†=∑α∈Ir,mjrdetjAA∗j.ai.∗αα∑α∈Ir,mAA∗αα.Remark 5.
For an arbitrary full-rank matrixA∈Hrm×n, a column vector b∈Hn×1 and a row vector c∈H1×m we put(11)cdetiA∗A.ib=∑β∈Jn,nicdetiA∗A.ibββ,detA∗A=∑β∈Jn,nA∗Aββwhenr=n,rdetjAA∗j.c=∑α∈Im,mjrdetjAA∗j.cαα,detAA∗=∑α∈Im,mAA∗ααwhenr=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 A∗∈Hrn×m determinantal representations of its Moore-Penrose inverse (A∗)†=(aij∗)†∈Hm×n are(12)aij∗†=aji†¯=∑α∈Ir,njrdetjA∗Aj.ai.αα∑α∈Ir,nA∗Aαα,(13)aij∗†=∑β∈Jr,micdetiAA∗.ia.jββ∑β∈Jr,mAA∗ββ.Since the projection matricesA†A≕PA=pij and AA†≕QA=qij are Hermitian, then pij=pji¯ and qij=qji¯ for all i≠j. So due to Theorem 4 and Remark 6 we have evidently the following corollaries.Corollary 7.
IfA∈Hrm×n, then the projection matrix PA=pijn×n has the determinantal representations(14)pij=∑β∈Jr,nicdetiA∗A.ia˙.jββ∑β∈Jr,nA∗Aββ=∑α∈Ir,njrdetjA∗Aj.a˙i.αα∑α∈Ir,nA∗Aαα, where a˙.j and a˙i. are the jth column and ith row of A∗A∈Hn×n, respectively.Corollary 8.
IfA∈Hrm×n, then the projection matrix AA†≕QA=qijm×m has the determinantal representation(15)qij=∑α∈Ir,mjrdetjAA∗j.a¨i.αα∑α∈Ir,mAA∗αα=∑β∈Jr,micdetiAA∗.ia¨.jββ∑α∈Jr,mAA∗ββ, where a¨i. and a¨j. are the ith row and the jth column of AA∗∈Hm×m.Determinantal representations of orthogonal projectorsLA≔I-A†A and RA≔I-AA† induced from A can be derived similarly.Theorem 9 (see [3]).
LetA∈Hm×n, B∈Hr×s, C∈Hm×s be known and X∈Hn×r be unknown. Then the matrix equation(16)AXB=Cis consistent if and only if AA†CB†B=C. In this case, its general solution can be expressed as(17)X=A†CB†+LAV+WRB,where V,W are arbitrary matrices over H with allowable dimensions.Theorem 10 (see [35]).
LetA∈Hr1m×n, B∈Hr2r×s. Then the partial solution X0=A†CB†=(xij0)∈Hn×r to (16) has determinantal representations,(18)xij0=∑β∈Jr1,nicdetiA∗A.id.jBββ∑β∈Jr1,nA∗Aββ∑α∈Ir2,rBB∗αα,or(19)xij0=∑α∈Ir2,rjrdetjBB∗j.di.Aαα∑β∈Jr1,nA∗Aββ∑α∈Ir2,rBB∗αα,where(20)d.jB=∑α∈Ir2,rjrdetjBB∗j.c~k.αα∈Hn×1,k=1,…,n,di.A=∑β∈Jr1,nicdetiA∗A.ic~.lββ∈H1×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~=A∗CB∗.Corollary 11.
LetA∈Hkm×n, C∈Hm×s be known and X∈Hn×s be unknown. Then the matrix equation AX=C is consistent if and only if AA†C=C. In this case, its general solution can be expressed as X=A†C+LAV, where V is an arbitrary matrix over H with an allowable dimension. The partial solution X0=A†C has the following determinantal representation,(21)xij0=∑β∈Jk,nicdetiA∗A.ic^.jββ∑β∈Jk,nA∗Aββ.where c^.j is the jth column of  C^=A∗C.Corollary 12.
LetB∈Hkr×s, C∈Hn×s be given and X∈Hn×r be unknown. Then the equation XB=C is solvable if and only if  C=CB†B 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=∑α∈Ik,rjrdetjBB∗j.c^i.αα∑α∈Ik,rBB∗αα. where c^i. is the ith row of  C^=CB∗.


### 2.3. Determinantal Representations of the General Solution to the Sylvester Matrix Equation (3)

Lemma 13 (see [3]).
LetA∈Hm×n, B∈Hr×s, C∈Hm×p, D∈Hq×s, E∈Hm×s. Put M=RAC, N=DLB, S=CLM. Then the following results are equivalent. (i)
Eq. (3) has a pair solution (X,Y), where X∈Hn×r, Y∈Hp×q.(ii)
RMRAE=0, RAELD=0, ELDLN=0, RCELB=0.(iii)
QMRAEPD=RAE, QCELBPN=ELB.(iv)
rankACE=rankAC, rankB∗D∗E∗=rankB∗D∗, rankAE0D=rankA00D, rankCE0B=rankC00B.
In that case, the general solution to (3) can be expressed as(23)X=A†EB†-A†CM†RAEB†-A†SC†ELBN†DB†--A†SVRNDB†+LAU+ZRB,(24)Y=M†RAED†+LMS†SC†ELBN†+LMV-S†SVNN†+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 [57]).
IfA∈Hn×n is Hermitian and idempotent, then for any matrix B∈Hm×n the following equations hold(25)ABA†=BA†,AB†A=AB†.SinceRA, LB, and LM are projectors, then by Lemma 14 the simplifications of (23) and (24) are as follows: (26)X=A†EB†-A†CM†EB†-A†SC†EN†DB†-A†SVRNDB†++LAU+ZRB,Y=M†ED†+PSC†EN†+LMV-PSVQN+WRD. By putting U,V,Z, and W as zero-matrices, we obtain the partial solution to (3),(27)X=A†EB†-A†CM†EB†-A†SC†EN†DB†,(28)Y=M†ED†+PSC†EN†.The following theorem gives determinantal representations of (27)-(28).Theorem 15 (see [45]).
LetA∈Hr1m×n, B∈Hr2r×s, C∈Hr3m×p, D∈Hr4q×s, rankM=r5, rankN=r6, rankS=r7. Then the pair solution (27)-(28), X=xij∈Hn×r, Y=xgf∈Hp×q, to (3) by the components (29)xij=xij1-xij2-xij3,ygf=ygf1+ygf2,has the determinantal representation, as follows.
(i)(30)xij1=∑β∈Jr1,nicdetiA∗A.id.jBββ∑β∈Jr1,nA∗Aββ∑α∈Ir2,rBB∗αα,or (31)xij1=∑α∈Ir2,rjrdetjBB∗j.di.Aαα∑β∈Jr1,nA∗Aββ∑α∈Ir2,rBB∗αα, where (32)d.jB=∑α∈Ir2,rjrdetjBB∗j.ek.1αα∈Hn×1,k=1,…,n,di.A=∑β∈Jr1,nicdetiA∗A.ie.l1ββ∈H1×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 E1≔A∗EB∗.
(ii)(33)xij2=∑t=1qφiq∑α∈Ir2,rjrdetjBB∗j.eq.2αα∑β∈Jr1,nA∗Aββ∑β∈Ir5,mMM∗αα∑α∈Ir2,rBB∗αα, where eq.(2) is qth row of E2≔EB∗. (34)φiq=∑β∈Jr1,nicdetiA∗A.iψ.qMββ=∑α∈Ir5,mqrdetqMM∗q.ψi.Aαα,ψ.qM=∑α∈Ir5,mqrdetqMM∗q.cf.1αα∈Hn×1,f=1,…,n,ψi.A=∑β∈Jr1,nicdetiA∗A.ic.s1ββ∈H1×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 C1≔A∗CM∗.
(iii)(35)xij3==∑t=1p∑f=1q∑β∈Jr1,nicdetiA∗A.is.t1ββ∑β∈Jr3,ptcdettC∗C.te.f3ββηfj∑β∈Jr1,nA∗Aββ∑β∈Jr3,pC∗Cββ∑α∈Jr6,sN∗Nββ∑α∈Ir2,rBB∗αα,where s.t(1) is the tth column of S1≔A∗S, e.f(3) is the fth column of E3≔C∗E, and (36)ηfj=∑α∈Ir2,rtrdetjBB∗j.ζf.Nαα=∑β∈Jr6,sfcdetfN∗N.fζ.jBββ,ζf.N=∑β∈Jr6,sfcdetfN∗N.fd.k1ββ∈H1×r,k=1,…,r,ζ.jB=∑α∈Ir2,rjrdetjBB∗j.dl.1αα∈Hs×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=N∗DB∗.
(iv)(37)ygf1=∑β∈Jr5,pgcdetgM∗M.gd.fDββ∑β∈Jr5,pM∗Mββ∑α∈Ir4,qDD∗αα,
or(38)ygf1=∑α∈Ir4,qfrdetfDD∗f.dg.Mαα∑β∈Jr5,pM∗Mββ∑α∈Ir4,qDD∗αα,
where(39)d.fD=∑α∈Ir4,qfrdetfDD∗f.ek.4αα∈Hp×1,k=1,…,p,dg.M=∑β∈Jr5,pgcdetgM∗M.ge.l4ββ∈H1×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 E4≔M∗ED∗.
(v)(40)ygf2=∑t=1p∑β∈Jr7,pgcdetgS∗S.gs¨.tββξtf∑β∈Jr7,pS∗Sββ∑β∈Jr3,pC∗Cββ∑α∈Ir6,qNN∗αα,where (41)ξtf=∑α∈Ir6,qtrdetfNN∗f.ϕt.Cαα=∑β∈Jr3,ptcdettC∗C.tϕ.fNββ,ϕt.C=∑β∈Jr3,ptcdettC∗C.te.k5ββ∈H1×q,k=1,…,q,ϕ.fN=∑α∈Ir6,qjrdetfNN∗f.el.5αα∈Hs×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=C∗EN∗.


## 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., [51–56]). 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.SupposeSn is the symmetric group on the set In={1,…,n}. Let A∈Hn×n. Row determinants of A along each row can be defined as follows.Definition 1 (see [32]).
The i th row determinant of A=(aij)∈Hn×n is defined for all i=1,…,n by putting(5)rdetiA=∑σ∈Sn-1n-raiik1aik1ik1+1…aik1+l1i…aikrikr+1…aikr+lrikr,σ=iik1ik1+1…ik1+l1ik2ik2+1…ik2+l2…ikrikr+1…ikr+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 [32]).
The j th column determinant of A=(aij)∈Hn×n is defined for all j=1,…,n by putting(6)cdetjA=∑τ∈Sn-1n-rajkrjkr+lr…ajkr+1jkr…ajjk1+l1…ajk1+1jk1ajk1j,τ=jkr+lr…jkr+1jkr…jk2+l2…jk2+1jk2jk1+l1…jk1+1jk1j,where jk2<jk3<⋯<jkr and jkt<jkt+s for t=2,…,r and s=1,…,lt.So an arbitraryn×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 [32],(7)rdet1A=⋯=rdetnA=cdet1A=⋯=cdetnA∈R,which enables definingthe determinant of a Hermitian matrix by putting(8)detA≔rdetiA=cdetiAfor all i=1,…,n.Its properties are similar to the properties of an usual (commutative) determinant and they have been completely explored in [32] by using row and column determinants that are so defined only by construction. We note the following that will be required below.Lemma 3.
LetA∈Hm×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.Letα≔α1,…,αk⊆1,…,m and β≔β1,…,βk⊆1,…,n be subsets of the order 1≤k≤minm,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 A∈Hn×n is Hermitian, then Aαα denotes the corresponding principal minor of detA.LetLk,n≔α:α=α1,…,αk,1≤α1<⋯<αk≤n denote the collection of strictly increasing sequences of k integers chosen from 1,…,n for all 1≤k≤n. Then, for fixed i∈α and j∈β, the collection of sequences of row indexes that contain the index i is denoted by Ir,mi≔α:α∈Lr,m,i∈α; similarly, the collection of sequences of column indexes that contain the index j is denote by Jr,nj≔β:β∈Lr,n,j∈β.Leta.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 [34]).
IfA∈Hrm×n, then the Moore-Penrose inverse A†=aij†∈Hn×m have the following determinantal representations,(9)aij†=∑β∈Jr,nicdetiA∗A.ia.j∗ββ∑β∈Jr,nA∗Aββ,(10)aij†=∑α∈Ir,mjrdetjAA∗j.ai.∗αα∑α∈Ir,mAA∗αα.Remark 5.
For an arbitrary full-rank matrixA∈Hrm×n, a column vector b∈Hn×1 and a row vector c∈H1×m we put(11)cdetiA∗A.ib=∑β∈Jn,nicdetiA∗A.ibββ,detA∗A=∑β∈Jn,nA∗Aββwhenr=n,rdetjAA∗j.c=∑α∈Im,mjrdetjAA∗j.cαα,detAA∗=∑α∈Im,mAA∗ααwhenr=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 A∗∈Hrn×m determinantal representations of its Moore-Penrose inverse (A∗)†=(aij∗)†∈Hm×n are(12)aij∗†=aji†¯=∑α∈Ir,njrdetjA∗Aj.ai.αα∑α∈Ir,nA∗Aαα,(13)aij∗†=∑β∈Jr,micdetiAA∗.ia.jββ∑β∈Jr,mAA∗ββ.Since the projection matricesA†A≕PA=pij and AA†≕QA=qij are Hermitian, then pij=pji¯ and qij=qji¯ for all i≠j. So due to Theorem 4 and Remark 6 we have evidently the following corollaries.Corollary 7.
IfA∈Hrm×n, then the projection matrix PA=pijn×n has the determinantal representations(14)pij=∑β∈Jr,nicdetiA∗A.ia˙.jββ∑β∈Jr,nA∗Aββ=∑α∈Ir,njrdetjA∗Aj.a˙i.αα∑α∈Ir,nA∗Aαα, where a˙.j and a˙i. are the jth column and ith row of A∗A∈Hn×n, respectively.Corollary 8.
IfA∈Hrm×n, then the projection matrix AA†≕QA=qijm×m has the determinantal representation(15)qij=∑α∈Ir,mjrdetjAA∗j.a¨i.αα∑α∈Ir,mAA∗αα=∑β∈Jr,micdetiAA∗.ia¨.jββ∑α∈Jr,mAA∗ββ, where a¨i. and a¨j. are the ith row and the jth column of AA∗∈Hm×m.Determinantal representations of orthogonal projectorsLA≔I-A†A and RA≔I-AA† induced from A can be derived similarly.Theorem 9 (see [3]).
LetA∈Hm×n, B∈Hr×s, C∈Hm×s be known and X∈Hn×r be unknown. Then the matrix equation(16)AXB=Cis consistent if and only if AA†CB†B=C. In this case, its general solution can be expressed as(17)X=A†CB†+LAV+WRB,where V,W are arbitrary matrices over H with allowable dimensions.Theorem 10 (see [35]).
LetA∈Hr1m×n, B∈Hr2r×s. Then the partial solution X0=A†CB†=(xij0)∈Hn×r to (16) has determinantal representations,(18)xij0=∑β∈Jr1,nicdetiA∗A.id.jBββ∑β∈Jr1,nA∗Aββ∑α∈Ir2,rBB∗αα,or(19)xij0=∑α∈Ir2,rjrdetjBB∗j.di.Aαα∑β∈Jr1,nA∗Aββ∑α∈Ir2,rBB∗αα,where(20)d.jB=∑α∈Ir2,rjrdetjBB∗j.c~k.αα∈Hn×1,k=1,…,n,di.A=∑β∈Jr1,nicdetiA∗A.ic~.lββ∈H1×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~=A∗CB∗.Corollary 11.
LetA∈Hkm×n, C∈Hm×s be known and X∈Hn×s be unknown. Then the matrix equation AX=C is consistent if and only if AA†C=C. In this case, its general solution can be expressed as X=A†C+LAV, where V is an arbitrary matrix over H with an allowable dimension. The partial solution X0=A†C has the following determinantal representation,(21)xij0=∑β∈Jk,nicdetiA∗A.ic^.jββ∑β∈Jk,nA∗Aββ.where c^.j is the jth column of  C^=A∗C.Corollary 12.
LetB∈Hkr×s, C∈Hn×s be given and X∈Hn×r be unknown. Then the equation XB=C is solvable if and only if  C=CB†B 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=∑α∈Ik,rjrdetjBB∗j.c^i.αα∑α∈Ik,rBB∗αα. where c^i. is the ith row of  C^=CB∗.


## 2.3. Determinantal Representations of the General Solution to the Sylvester Matrix Equation (3)

Lemma 13 (see [3]).
LetA∈Hm×n, B∈Hr×s, C∈Hm×p, D∈Hq×s, E∈Hm×s. Put M=RAC, N=DLB, S=CLM. Then the following results are equivalent. (i)
Eq. (3) has a pair solution (X,Y), where X∈Hn×r, Y∈Hp×q.(ii)
RMRAE=0, RAELD=0, ELDLN=0, RCELB=0.(iii)
QMRAEPD=RAE, QCELBPN=ELB.(iv)
rankACE=rankAC, rankB∗D∗E∗=rankB∗D∗, rankAE0D=rankA00D, rankCE0B=rankC00B.
In that case, the general solution to (3) can be expressed as(23)X=A†EB†-A†CM†RAEB†-A†SC†ELBN†DB†--A†SVRNDB†+LAU+ZRB,(24)Y=M†RAED†+LMS†SC†ELBN†+LMV-S†SVNN†+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 [57]).
IfA∈Hn×n is Hermitian and idempotent, then for any matrix B∈Hm×n the following equations hold(25)ABA†=BA†,AB†A=AB†.SinceRA, LB, and LM are projectors, then by Lemma 14 the simplifications of (23) and (24) are as follows: (26)X=A†EB†-A†CM†EB†-A†SC†EN†DB†-A†SVRNDB†++LAU+ZRB,Y=M†ED†+PSC†EN†+LMV-PSVQN+WRD. By putting U,V,Z, and W as zero-matrices, we obtain the partial solution to (3),(27)X=A†EB†-A†CM†EB†-A†SC†EN†DB†,(28)Y=M†ED†+PSC†EN†.The following theorem gives determinantal representations of (27)-(28).Theorem 15 (see [45]).
LetA∈Hr1m×n, B∈Hr2r×s, C∈Hr3m×p, D∈Hr4q×s, rankM=r5, rankN=r6, rankS=r7. Then the pair solution (27)-(28), X=xij∈Hn×r, Y=xgf∈Hp×q, to (3) by the components (29)xij=xij1-xij2-xij3,ygf=ygf1+ygf2,has the determinantal representation, as follows.
(i)(30)xij1=∑β∈Jr1,nicdetiA∗A.id.jBββ∑β∈Jr1,nA∗Aββ∑α∈Ir2,rBB∗αα,or (31)xij1=∑α∈Ir2,rjrdetjBB∗j.di.Aαα∑β∈Jr1,nA∗Aββ∑α∈Ir2,rBB∗αα, where (32)d.jB=∑α∈Ir2,rjrdetjBB∗j.ek.1αα∈Hn×1,k=1,…,n,di.A=∑β∈Jr1,nicdetiA∗A.ie.l1ββ∈H1×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 E1≔A∗EB∗.
(ii)(33)xij2=∑t=1qφiq∑α∈Ir2,rjrdetjBB∗j.eq.2αα∑β∈Jr1,nA∗Aββ∑β∈Ir5,mMM∗αα∑α∈Ir2,rBB∗αα, where eq.(2) is qth row of E2≔EB∗. (34)φiq=∑β∈Jr1,nicdetiA∗A.iψ.qMββ=∑α∈Ir5,mqrdetqMM∗q.ψi.Aαα,ψ.qM=∑α∈Ir5,mqrdetqMM∗q.cf.1αα∈Hn×1,f=1,…,n,ψi.A=∑β∈Jr1,nicdetiA∗A.ic.s1ββ∈H1×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 C1≔A∗CM∗.
(iii)(35)xij3==∑t=1p∑f=1q∑β∈Jr1,nicdetiA∗A.is.t1ββ∑β∈Jr3,ptcdettC∗C.te.f3ββηfj∑β∈Jr1,nA∗Aββ∑β∈Jr3,pC∗Cββ∑α∈Jr6,sN∗Nββ∑α∈Ir2,rBB∗αα,where s.t(1) is the tth column of S1≔A∗S, e.f(3) is the fth column of E3≔C∗E, and (36)ηfj=∑α∈Ir2,rtrdetjBB∗j.ζf.Nαα=∑β∈Jr6,sfcdetfN∗N.fζ.jBββ,ζf.N=∑β∈Jr6,sfcdetfN∗N.fd.k1ββ∈H1×r,k=1,…,r,ζ.jB=∑α∈Ir2,rjrdetjBB∗j.dl.1αα∈Hs×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=N∗DB∗.
(iv)(37)ygf1=∑β∈Jr5,pgcdetgM∗M.gd.fDββ∑β∈Jr5,pM∗Mββ∑α∈Ir4,qDD∗αα,
or(38)ygf1=∑α∈Ir4,qfrdetfDD∗f.dg.Mαα∑β∈Jr5,pM∗Mββ∑α∈Ir4,qDD∗αα,
where(39)d.fD=∑α∈Ir4,qfrdetfDD∗f.ek.4αα∈Hp×1,k=1,…,p,dg.M=∑β∈Jr5,pgcdetgM∗M.ge.l4ββ∈H1×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 E4≔M∗ED∗.
(v)(40)ygf2=∑t=1p∑β∈Jr7,pgcdetgS∗S.gs¨.tββξtf∑β∈Jr7,pS∗Sββ∑β∈Jr3,pC∗Cββ∑α∈Ir6,qNN∗αα,where (41)ξtf=∑α∈Ir6,qtrdetfNN∗f.ϕt.Cαα=∑β∈Jr3,ptcdettC∗C.tϕ.fNββ,ϕt.C=∑β∈Jr3,ptcdettC∗C.te.k5ββ∈H1×q,k=1,…,q,ϕ.fN=∑α∈Ir6,qjrdetfNN∗f.el.5αα∈Hs×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=C∗EN∗.


## 3. Determinantal Representations of the General and (Skew-)Hermitian Solutions to (4)

Now consider (4). Since for an arbitrary matrix A it is evident that PA∗=A∗†A∗=AA†∗=QA, so QA∗=PA, LA∗=I-PA∗=I-QA=RA, and RA∗=LA. Due to the above, M=RAB and N=B∗LA∗=B∗RA=(RAB)∗=M∗, and we obtain the following analog of Lemma 13.Lemma 16.
LetA∈Hm×n, B∈Hm×k, C∈Hm×m. Put M=RAB, S=BLM. Then the following results are equivalent.(i)
Equation (4) has a pair solution (X,Y), where X∈Hn×n, Y∈Hk×k.(ii)
RMRAC=0, RACRB=0, CRBRM=0, RBCRA=0.(iii)
QMRACQB=RAC, QBCRAQM=CRA.(iv)
rankABC=rankAB, rankAC0B∗=rankA00B∗, rankBC0A∗=rankB00A∗.
In that case, the general solution to (4) can be expressed as follows: (42)X=A†CA∗,†-A†BM†CA∗,†-A†SB∗,†CM∗,†B∗A∗,†-A†SVLMB∗A∗,†+LAU+ZLA,Y=M†CB∗,†+PSB†CM∗,†+LMV-PSVPM+WLB. where U, V, Z, and W are arbitrary matrices over H with allowable dimensions.By puttingU, V, Z, and W as zero-matrices with compatible dimensions, we obtain the following partial solution to (4),(43)X=A†CA∗,†-A†BM†CA∗,†-A†SB†CM∗,†B∗A∗,†,(44)Y=M†CB∗,†+PSB†CM∗,†.The following theorem gives determinantal representations of (43)-(44).Theorem 17.
LetA∈Hr1m×n, B∈Hr2m×k, rankM=r3, rankS=r4. Then the partial pair solution (43)-(44) to (4), X=xij∈Hn×n, Y=ypg∈Hk×k, by the components(45)xij=xij1-xij2-xij3,ypg=ypg1+ypg2,possesses the following determinantal representations:
(i)(46)xij1=∑α∈Ir1,njrdetjA∗Aj.vi.αα∑α∈Ir1,nA∗Aαα2or(47)xij1=∑β∈Jr1,nicdetiA∗A.iv.jββ∑β∈Jr1,nA∗Aββ2,where(48)vi.=∑β∈Jr1,nicdetiA∗A.ic.s1ββ∈H1×n,s=1,…,n,(49)v.j=∑α∈Ir1,njrdetjA∗Aj.cf.1αα∈Hn×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=A∗CA.
(ii)(50)xij2=∑α∈Ir1,njrdetjA∗Aj.ϕ~i.αα∑β∈Jr1,nA∗Aββ2∑α∈Ir3,mMM∗αα,where ϕ~i. is the ith row of Φ~≔ΦCA and Φ=(ϕiq)∈Hn×m is such that(51)ϕiq=∑β∈Jr1,nicdetiA∗A.iη.qMββ=∑α∈Ir3,mqrdetqMM∗q.ηi.Aαα,(52)η.qM=∑α∈Ir3,mqrdetqMM∗q.bf.1αα∈Hn×1,f=1,…,n,ηi.A=∑β∈Jr1,nicdetiA∗A.ib.s1ββ∈H1×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=A∗BM∗ and cq.(2) is the qth row of C2=CA.
(iii)(53)xij3=∑β∈Jr1,nicdetiA∗A.iυ~.jββ∑β∈Jr1,nA∗Aββ2∑β∈Jr2,kB∗Bββ∑β∈Jr3,mMM∗ββ,where υ~.j is the jth column of Υ~=A∗SΥ, the matrix Υ=(υpj)∈Hk×n such that (54)υpj=∑β∈Jr2,kpcdetpB∗B.pc~.jββ, where c~.j is the jth column of C~=B∗CΦ∗ and Φ∗ is Hermitian adjoint to Φ=(ϕiq) from (51).
(iv)(55)ypg1=∑β∈Jr3,kpcdetpM∗M.pd.gBββ∑β∈Jr3,kM∗Mββ∑α∈Ir2,kB∗Bαα,or(56)ypg1=∑α∈Ir2,kgrdetgB∗Bg.dp.Mαα∑β∈Jr3,kM∗Mββ∑α∈Ir2,kB∗Bαα,where (57)d.gB=∑α∈Ir2,kgrdetgB∗Bg.cq.4αα∈Hk×1,q=1,…,k,dp.M=∑β∈Jr3,kpcdetpM∗M.pc.l4ββ∈H1×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 C4≔M∗CB.
(v)(58)ypg2=∑β∈Jr4,kpcdetpS∗S.pω~.gββ∑β∈Jr4,kS∗Sββ∑β∈Jr2,kB∗Bββ∑α∈Ir3,kM∗Mαα,where Ω~=S∗SΩ and Ω=(ωtg) such that(59)ωtg=∑β∈Jr2,ktcdettB∗B.td.gMββ=∑α∈Ir3,kgrdetgM∗Mg.dt.Bαα,d.gM=∑α∈Ir3,kgrdetgM∗Mg.cq.4,∗αα∈Hk×1,q=1,…,k,dt.B=∑β∈Jr3,ktcdettB∗B.tc.l4,∗ββ∈H1×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 C4∗≔M∗CB.Proof.
The proof evidently follows from the proof of Theorem15 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=A†CA∗†=(xij(1)), we have (60)xij1=∑l=1m∑t=1mail†cltatj∗,†. By using determinantal representations (9) and (13) of the Moore-Penrose inverses A† and A∗†, respectively, we obtain (61)xij1=∑l=1m∑t=1m∑β∈Jr1,nicdetiA∗A.ia.l∗ββclt∑α∈Ir1,njrdetjA∗Aj.at.αα∑α∈Ir1,nA∗Aαα∑β∈Jr1,nA∗Aββ. 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 C1≔A∗CA. Since ∑l=1m∑t=1mafl∗cltats=cfs(1), then (62)xij1=∑f=1n∑s=1n∑β∈Jr1,nicdetiA∗A.ie.fββcfs1∑α∈Ir1,njrdetjA∗Aj.es.αα∑α∈Ir1,nA∗Aαα∑β∈Jr1,nA∗Aββ If we denote by (63)vis≔∑f=1n∑β∈Jr1,nicdetiA∗A.ie.fββcfs1=∑β∈Jr1,nicdetiA∗A.ic.s1ββ the sth component of a row vector vi.=vi1,…,vin, then (64)∑s=1mvis∑α∈Ir1,njrdetjA∗Aj.es.αα=∑α∈Ir1,njrdetjA∗Aj.vi.αα. Further, it is evident that ∑β∈Jr1,nA∗Aββ=∑α∈Ir1,nA∗Aαα, so the first term of (43) has the determinantal representation (46), where vi. is (48).
If we denote by(65)vfj2≔∑s=1ncfs1∑α∈Ir1,njrdetjA∗Aj.es.αα==∑α∈Ir1,njrdetjA∗Aj.cf.1ααthe fth component of a column vector v.j=v1j,…,vnj, then(66)∑f=1n∑β∈Jr1,nicdetiA∗A.ie.fββvfj=∑β∈Jr1,nicdetiA∗A.iv.jββ. So another determinantal representation of the first term of (43) is (47), where v.j is (49).
(ii) For the second termA†BM†CA∗,†≔X2=xij(2) of (43), we have (67)xij2=∑l=1m∑p=1k∑q=1m∑t=1mail†blpmpq†cqtatj∗,†. Using determinantal representations (9) for the Moore-Penrose inverse A†, (10) for M†=(mpq†), and (13) for A∗†, respectively, we obtain (68)xij2=∑l=1m∑p=1k∑q=1m∑t=1m∑β∈Jr1,nicdetiA∗A.ia.l∗ββblp∑α∈Ir3,mqrdetqMM∗q.mp.∗αα∑β∈Jr1,nA∗Aββ∣∑α∈Ir3,mMM∗αα××cqt∑α∈Ir1,njrdetjA∗Aj.at.αα∑α∈Ir1,nA∗Aαα.Further, thinking as above in the point (i), we obtain(69)ϕiq≔∑l=1m∑p=1k∑β∈Jr1,nicdetiA∗A.ia.l∗ββblp·∑α∈Ir3,mqrdetqMM∗q.mp.∗αα=∑β∈Jr1,nicdetiA∗A.iη.qMββ=∑α∈Ir3,mqrdetqMM∗q.ηi.Aαα,where (70)η.qM=∑α∈Ir3,mqrdetqMM∗q.bf.1αα∈Hn×1,f=1,…,n,ηi.A=∑β∈Jr1,nicdetiA∗A.ib.s1ββ∈H1×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=A∗BM∗. Construct the matrix Φ=(ϕiq)∈Hn×m such that ϕiq are obtained by (69), and denote Φ~≔ΦCA. Since(71)∑q=1m∑t=1mϕiqcqt∑α∈Ir1,njrdetjA∗Aj.at.αα=∑α∈Ir1,njrdetjA∗Aj.ϕ~i.αα, where ϕ~i. is the ith row of Φ~, then we have (50).
(iii) For the third termA†SB†CM∗,†B∗A∗,†≔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∗,†B∗A∗,†=(A†BM†)∗, we have (72)xij3==∑p=1k∑t=1m∑β∈Jr1,nicdetiA∗A.is.p1ββ∑β∈Jr2,kpcdetpB∗B.pc.t3ββϕtj∗∑β∈Jr1,nA∗Aββ2∑β∈Jr2,kB∗Bββ∑β∈Jr3,mMM∗ββ, where s.p(1) is the pth column of S1≔A∗S, c.t(3) is the tth column of C3≔B∗C, ϕtj∗ is the tjth entry of Φ∗ that is Hermitian adjoint to Φ=(ϕiq) from (51). Denote C3Φ∗=B∗CΦ∗=C~. Then,(73)∑t=1m∑β∈Jr2,kpcdetpB∗B.pc.t3ββϕtj∗=∑β∈Jr2,kpcdetpB∗B.pc~.jββ.Construct the matrix Υ=(υpj)∈Hk×n such that (74)υpj=∑β∈Jr2,kpcdetpB∗B.pc~.jββ. Denote S1Υ=A∗SΥ≕Υ~=(υ~ij)∈Hn×n. Since(75)∑β∈Jr1,nicdetiA∗A.is.p1ββυpj=∑β∈Jr1,nicdetiA∗A.iυ~.jββ, it follows (53).
(iv) Due to Theorem10 and similarly as above for the first term Y1=M†CB∗,†=(ypg(1)) of (44), we have the determinantal representations (55) and (56).
(v) Finally, for the second termY2=PSB†CM∗,†=ypg(2) of (44) using (14) for a determinantal representation of PS, and due to Theorem 10 for B†CM∗,†, we obtain (76)ypg2=∑t=1k∑β∈Jr4,kpcdetpS∗S.ps¨.tββωtg∑β∈Jr4,kS∗Sββ∑β∈Jr2,kB∗Bββ∑α∈Ir3,kM∗Mαα, where ϕtg are(77)ωtg=∑β∈Jr2,ktcdettB∗B.td.gMββ=∑α∈Ir3,kgrdetgM∗Mg.dt.Bαα,(78)d.gM=∑α∈Ir3,kgrdetgM∗Mg.cq.4,∗αα∈Hk×1,q=1,…,k,dt.B=∑β∈Jr3,ktcdettB∗B.tc.l4,∗ββ∈H1×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 C4∗≔M∗CB. Construct the matrix Ω=(ωtg)∈Hk×k such that ωtg are obtained by (77), and denote Ω~≔S∗SΩ. Since(79)∑t=1k∑β∈Jr4,kpcdetpS∗S.ps¨.tββωtg=∑β∈Jr4,kpcdetpS∗S.pω~.gββ,it follows (58).Due to [24], the following lemma can be generalized to H.Lemma 18.
Suppose that matricesA∈Hm×n and B∈Hm×m and C∈Hm×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∗=A†CA∗,†-A†CM∗,†B∗A∗,†-A†BM†CB∗,†S∗A∗,†,Y∗=B†CM∗,†+M†CB∗,†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¯=∑β∈Jr1,njcdetiA∗A.iϕ~.j∗ββ∑β∈Jr1,nA∗Aββ2∑β∈Jr3,kM∗Mββ, where ϕ~.j∗ is the jth column of Φ~∗=A∗CΦ∗ that is Hermitian adjoint to Φ~ from the point (i) of Theorem 17, and Φ∗=(ϕqj∗)∈Hn×m is such that(83)ϕqj∗=∑α∈Ir1,njrdetjA∗Aj.ζq.Mαα=∑β∈Jr3,mqcdetqMM∗.qζ.jAββ,(84)ζq.M=∑β∈Jr3,mqcdetqMM∗.qb.f1,∗ββ∈H1×n,f=1,…,n,ζ.jA=∑α∈Ir1,njrdetjA∗Aj.bs.1,∗αα∈H1×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∗=MB∗A.(iii)(85)xji3¯=∑α∈Ir1,njrdetjA∗Aj.υ~i.∗αα∑α∈Ir1,nA∗Aαα2∑β∈Ir2,kB∗Bαα∑β∈Ir3,mMM∗αα, where υ~i.∗ is the ith row of Υ~∗=Υ∗S∗A, the matrix Υ∗=(υip∗)∈Hn×k such that (86)υip∗=∑α∈Ir2,kprdetpB∗Bp.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¯=∑α∈Ir3,kgrdetgM∗Mg.dp.Bαα∑α∈Ir3,kM∗Mαα∑β∈Jr2,kB∗Bββ=∑β∈Jr2,kpcdetpB∗B.pd.gMββ∑α∈Ir3,kM∗Mαα∑β∈Jr2,kB∗Bββ, where (88)dp.B=∑β∈Jr2,kpcdetpB∗B.pc.q4,∗ββ∈H1×k,q=1,…,k,d.gM=∑α∈Ir3,kgrdetgM∗Mg.cl.4,∗αα∈Hk×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 C4∗≔B∗CM.(ii)(89)ygp2¯=∑α∈Ir4,kprdetpS∗Sp.ω~g.∗αα∑α∈Ir4,kS∗Sαα∑β∈Jr2,mB∗Bββ∑α∈Ir3,mMM∗αα, where Ω~∗=Ω∗S∗S and Ω∗=(ωgt∗) such that(90)ωgt∗=∑β∈Jr3,kgcdetgM∗M.gd.tBββ=∑α∈Ir2,ktrdettB∗Bt.dg.Mαα,d.tB=∑α∈Ir2,ktrdettB∗Bt.cq.4αα∈H1×k,q=1,…,k,dg.M=∑β∈Jr3,kpcdetgM∗M.gc.l4ββ∈Hk×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 C4≔M∗CB.Remark 19.
By Lemma18, 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 detA∗A=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=A†CA∗,†-A†BM†CA∗,†-A†SB†CM∗,†B∗A∗,†=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=M†CB∗,†+PSB†CM∗,†=(1/63)-4+8j+10k.Now, we find the solution to (91) by determinantal representations. So, (95)MM∗=941j-j1,C2≔CA=2-1+ij+kj+k1-i,B1=A∗BM∗=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 ∑β∈J1,2A∗Aββ2=64, ∑α∈I1,2MM∗αα=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 allxij, 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 equationAXA∗+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.



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