Cramer's rules for the solution to the two-sided restricted quaternion matrix equation
aa r X i v : . [ m a t h . R A ] A ug Cramer’s rules for the solution to the two-sided restrictedquaternion matrix equation.
Ivan I. Kyrchei Abstract
Weighted singular value decomposition (WSVD) of a quaternion ma-trix and with its help determinantal representations of the quaternionweighted Moore-Penrose inverse have been derived recently by the au-thor. In this paper, using these determinantal representations, explicitdeterminantal representation formulas for the solution of the restrictedquaternion matrix equations,
AXB = D , and consequently, AX = D and XB = D are obtained within the framework of the theory of column-row determinants. We consider all possible cases depending on weightedmatrices. Keywords
Weighted singular value decomposition, Weighted Moore-Penroseinverse, Quaternion matrix, Matrix equation, Cramer rule
Mathematics subject classifications
Let R and C be the real and complex number fields, respectively. Throughoutthe paper, we denote the set of all m × n matrices over the quaternion skew field H = { a + a i + a j + a k | i = j = k = − , a , a , a , a ∈ R } by H m × n , and by H m × nr the set of all m × n matrices over H with a rank r .Let M ( n, H ) be the ring of n × n quaternion matrices and I be the identitymatrix with the appropriate size. For A ∈ H n × m , we denote by A ∗ , rank A theconjugate transpose (Hermitian adjoint) matrix and the rank of A . The matrix A = ( a ij ) ∈ H n × n is Hermitian if A ∗ = A .The definitions of the Moore-Penrose inverse [1] and the weighted Moore-Penrose inverse [2] can be extended to quaternion matrices as follows.The Moore-Penrose inverse of A ∈ H m × n , denoted by A † , is the uniquematrix X ∈ H n × m satisfying the following equations [1], AXA = A ; (1) XAX = X ; (2)( AX ) ∗ = AX ; (3)( XA ) ∗ = XA . (4)Let Hermitian positive definite matrices M and N of order m and n , respec-tively, be given. For A ∈ H m × n , the weighted Moore-Penrose inverse of [email protected],Pidstrygach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine,Ukraine is the unique solution X = A † M,N of the matrix equations (1) and (2) andthe following equations in X [3]:(3 M ) ( MAX ) ∗ = MAX ; (4 N ) ( NXA ) ∗ = NXA . In particular, when M = I m and N = I n , the matrix X satisfying the equations(1), (2), (3M), (4N) is the Moore-Penrose inverse A † .A basic method for finding the Moore-Penrose inverse is based on the singu-lar value decomposition (SVD). It is also available for quaternion matrices, (see,e.g. [4,5]). The weighted Moore-Penrose inverse A † M,N ∈ C m × n (over complex orreal fields) has the explicit expressing by the weighted singular value decompo-sition (WSVD) that at first has been obtained in [6] by Cholesky factorization.In [7], WSVD of real matrices with singular weights has been derived usingweighted orthogonal matrices and weighted pseudoorthogonal matrices.Recently, by the author, WSVD has been expanded to quaternion matrices. Theorem 1.1 [8] Let A ∈ H m × nr , M and N be positive definite matrices oforder m and n , respectively. Denote A ♯ = N − A ∗ M . There exist U ∈ H m × m , V ∈ H n × n satisfying U ∗ MU = I m and V ∗ N − V = I n such that A = UDV ∗ ,where D = (cid:18) Σ 00 0 (cid:19) . Then the weighted Moore-Penrose inverse A † M,N canbe represented A † M,N = N − V (cid:18) Σ −
00 0 (cid:19) U ∗ M , (5) where Σ = diag ( σ , σ , ..., σ r ) , σ ≥ σ ≥ ... ≥ σ r > and σ i is the nonzeroeigenvalues of A ♯ A or AA ♯ , which coincide. By using WSVD, within the framework of the theory of column-row deter-minants, limit and determinantal representations of the quaternion weightedMoore-Penrose inverse has been derived ibidem as well.But why determinantal representations of generalized inverses are so impor-tant? When we return to the usual inverse, its determinantal representation isthe matrix with cofactors in entries that gives direct method of its finding andmakes it applicable in Cramer’s rule for systems of linear equations. The samebe wanted for generalized inverses. But there is not so unambiguous even forcomplex or real matrices. Therefore, there are various determinantal represen-tations of generalized inverses because searches of their explicit more applicableexpressions are continuing (see, e.g. [9–13]).The understanding of the problem for determinantal representing of general-ized inverses as well as solutions and generalized inverse solutions of quaternionmatrix equations, only now begins to be decided due to the theory of column-rowdeterminants introduced in [14, 15].Song at al. [16, 17] have studied the weighted Moore-Penrose inverse overthe quaternion skew field and obtained its determinantal representation withinthe framework of the theory of column-row determinants as well. But WSVDof quaternion matrices has not been considered and for obtaining a determi-nantal representation there was used auxiliary matrices which different from A ,2nd weights M and N . Despite this in [17], Cramer’s rule of the quaternionrestricted matrix equation AXB = D has been derived with the help obtaineddeterminantal representations of the weighted Moore-Penrose inverse.The main goals of the paper are obtaining Cramer’s rule for the quaternionrestricted matrix equation AXB = D , and consequently, AX = D and XB = D using the determinantal representations of the weighted Moore-Penrose in-verse obtained by WSVD in [8]. We consider all possible cases with respect toweights of A and B .It need to note that currently the theory of column-row determinants ofquaternion matrices is active developing. Within the framework of column-row determinants, determinantal representations of various kind of generalizedinverses, (generalized inverses) solutions of quaternion matrix equations recentlyhave been derived as by the author (see, e.g. [18–22]) so by other researchers(see, e.g. [23–26]).In this chapter we shall adopt the following notation.Let α := { α , . . . , α k } ⊆ { , . . . , m } and β := { β , . . . , β k } ⊆ { , . . . , n } besubsets of the order 1 ≤ k ≤ min { m, n } . By A αβ denote the submatrix of A determined by the rows indexed by α , and the columns indexed by β . Then, A αα denotes a principal submatrix determined by the rows and columns indexedby α . If A ∈ M ( n, H ) is Hermitian, then by | A αα | denote the correspondingprincipal minor of det A , since A αα is Hermitian as well. For 1 ≤ k ≤ n ,denote by L k,n := { α : α = ( α , . . . , α k ) , ≤ α ≤ . . . ≤ α k ≤ n } the collectionof strictly increasing sequences of k integers chosen from { , . . . , n } . For fixed i ∈ α and j ∈ β , let I r, m { i } := { α : α ∈ L r,m , i ∈ α } , J r, n { j } := { β : β ∈ L r,n , j ∈ β } . The paper is organized as follows. We start with some basic concepts andresults from the theories of row-column determinants and of quaternion matricesin Section 2. Cramer’s rules for the quaternionic restricted matrix equation
AXB = D , and consequently, AX = D and XB = D are derived in Section3. All possible cases are considered in the three subsections of Section 3. InSection 4, we give numerical an example to illustrate the main results. For a quadratic matrix A = ( a ij ) ∈ M ( n, H ) can be define n row determinantsand n column determinants as follows.Suppose S n is the symmetric group on the set I n = { , . . . , n } . Definition 2.1 [14] The i th row determinant of A = ( a ij ) ∈ M ( n, H ) is efined for all i = 1 , . . . , n by putting rdet i A = X σ ∈ S n ( − n − r ( a ii k a i k i k . . .a i k l i ) . . . ( a i kr i kr +1 . . . a i kr + lr i kr ) ,σ = ( i i k i k +1 . . . i k + l ) ( i k i k +1 . . . i k + l ) . . . ( i k r i k r +1 . . . i k r + l r ) , with conditions i k < i k < . . . < i k r and i k t < i k t + s for t = 2 , . . . , r and s = 1 , . . . , l t . Definition 2.2 [14] The j th column determinant of A = ( a ij ) ∈ M ( n, H ) isdefined for all j = 1 , . . . , n by putting cdet j A = X τ ∈ S n ( − n − r ( a j kr j kr + lr . . . a j kr +1 i kr ) . . . ( a j j k l . . . a j k j k a j k j ) ,τ = ( j k r + l r . . . j k r +1 j k r ) . . . ( j k + l . . . j k +1 j k ) ( j k + l . . . j k +1 j k j ) , with conditions, j k < j k < . . . < j k r and j k t < j k t + s for t = 2 , . . . , r and s = 1 , . . . , l t . Suppose A ij denotes the submatrix of A obtained by deleting both the i throw and the j th column. Let a .j be the j th column and a i. be the i th row of A . Suppose A .j ( b ) denotes the matrix obtained from A by replacing its j thcolumn with the column b , and A i. ( b ) denotes the matrix obtained from A byreplacing its i th row with the row b . We note some properties of column androw determinants of a quaternion matrix A = ( a ij ), where i ∈ I n , j ∈ J n and I n = J n = { , . . . , n } . Proposition 2.1 [14] If b ∈ H , then rdet i A i. ( b · a i. ) = b · rdet i A , cdet i A .i ( a .i · b ) = cdet i A · b, for all i = 1 , . . . , n . Proposition 2.2 [14] If for A ∈ M ( n, H ) there exists t ∈ I n such that a tj = b j + c j for all j = 1 , . . . , n , then rdet i A = rdet i A t. ( b ) + rdet i A t. ( c ) , cdet i A = cdet i A t. ( b ) + cdet i A t. ( c ) , where b = ( b , . . . , b n ) , c = ( c , . . . , c n ) and for all i = 1 , . . . , n . Proposition 2.3 [14] If for A ∈ M ( n, H ) there exists t ∈ J n such that a i t = b i + c i for all i = 1 , . . . , n , then rdet j A = rdet j A .t ( b ) + rdet j A .t ( c ) , cdet j A = cdet j A .t ( b ) + cdet j A .t ( c ) , where b = ( b , . . . , b n ) T , c = ( c , . . . , c n ) T and for all j = 1 , . . . , n . emark 2.1 Let rdet i A = n P j =1 a ij · R ij and cdet j A = n P i =1 L ij · a ij for all i, j = 1 , . . . , n , where by R ij and L ij denote the right and left ( ij ) th cofactorsof A ∈ M ( n, H ) , respectively. It means that rdet i A can be expand by rightcofactors along the i th row and cdet j A can be expand by left cofactors along the j th column, respectively, for all i, j = 1 , . . . , n . The main property of the usual determinant is that the determinant of a non-invertible matrix must be equal zero. But the row and column determinantsdon’t satisfy it, in general. Therefore, these matrix functions can be consideras some pre-determinants. The following theorem has a key value in the theoryof the column and row determinants.
Theorem 2.1 [14] If A = ( a ij ) ∈ M ( n, H ) is Hermitian, then rdet A = · · · =rdet n A = cdet A = · · · = cdet n A ∈ R . Due to Theorem 2.1, we can define the determinant of a Hermitian matrix A ∈ M ( n, H ) by putting, det A := rdet i A = cdet i A , for all i = 1 , . . . , n . By usingits row and column determinants, the determinant of a quaternion Hermitianmatrix has properties similar to the usual determinant. These properties arecompletely explored in [14,15] and can be summarized in the following theorems. Theorem 2.2
If the i th row of a Hermitian matrix A ∈ M ( n, H ) is replacedwith a left linear combination of its other rows, i.e. a i. = c a i . + . . . + c k a i k . ,where c l ∈ H for all l = 1 , . . . , k and { i, i l } ⊂ I n , then rdet i A i . ( c a i . + . . . + c k a i k . ) = cdet i A i . ( c a i . + . . . + c k a i k . ) = 0 . Theorem 2.3
If the j th column of a Hermitian matrix A ∈ M ( n, H ) is replacedwith a right linear combination of its other columns, i.e. a .j = a .j c + . . . + a .j k c k , where c l ∈ H for all l = 1 , . . . , k and { j, j l } ⊂ J n , then cdet j A .j ( a .j c + . . . + a .j k c k ) = rdet j A .j ( a .j c + . . . + a .j k c k ) = 0 . The following theorem about determinantal representation of an inverse matrixof Hermitian follows immediately from these properties.
Theorem 2.4 [15] If a Hermitian matrix A ∈ M ( n, H ) is such that det A = 0 ,then there exist a unique right inverse matrix ( R A ) − and a unique left inversematrix ( L A ) − , and ( R A ) − = ( L A ) − =: A − , which possess the followingdeterminantal representations: ( R A ) − = 1det A R R · · · R n R R · · · R n · · · · · · · · · · · · R n R n · · · R nn , (6)( L A ) − = 1det A L L · · · L n L L · · · L n · · · · · · · · · · · · L n L n · · · L nn , (7)5 here det A = n P j =1 a ij · R ij = n P i =1 L ij · a ij , R ij = (cid:26) − rdet j A ii.j ( a . i ) , i = j, rdet k A ii , i = j, L ij = (cid:26) − cdet i A j ji. ( a j . ) , i = j, cdet k A j j , i = j. The submatrix A ii.j ( a . i ) is obtained from A by replacing the j th column with the i th column and then deleting both the i th row and column, A jji. ( a j . ) is obtainedby replacing the i th row with the j th row, and then by deleting both the j throw and column, respectively. I n = { , . . . , n } , k = min { I n \ { i }} , for all i, j = 1 , . . . , n . Theorem 2.5 If A ∈ M ( n, H ) , then det AA ∗ = det A ∗ A . Definition 2.3
For A ∈ M ( n, H ) , the double determinant of A is defined byputting, ddet A := det AA ∗ = det A ∗ A . For arbitrary A ∈ M( n, H ), we have the following theorem on determinantalrepresentations of its inverse. Theorem 2.6
The necessary and sufficient condition of invertibility of A ∈ M( n, H ) is ddet A = 0 . Then there exists A − = ( L A ) − = ( R A ) − , where ( L A ) − = ( A ∗ A ) − A ∗ = 1ddet A L L . . . L n L L . . . L n . . . . . . . . . . . . L n L n . . . L nn ( R A ) − = A ∗ ( AA ∗ ) − = 1ddet A ∗ R R . . . R n R R . . . R n . . . . . . . . . . . . R n R n . . . R nn and L ij = cdet j ( A ∗ A ) .j ( a ∗ .i ) , R ij = rdet i ( AA ∗ ) i. (cid:0) a ∗ j. (cid:1) , for all i, j = 1 , . . . , n. Moreover, the following criterion of invertibility of a quaternion matrix can beobtained.
Theorem 2.7 If A ∈ M ( n, H ) , then the following statements are equivalent.i) A is invertible, i.e. A ∈ GL ( n, H ) ; ii) rows of A are left-linearly independent;iii) columns of A are right-linearly independent;iv) ddet A = 0 . .2 Some provisions of quaternion eigenvalues Due to real-scalar multiplying on the right, quaternion column-vectors forma right vector R -space, and, by real-scalar multiplying on the left, quaternionrow-vectors form a left vector R -space denoted by H r and H l , respectively. Itcan be shown that H r and H l possess corresponding H -valued inner productsby putting h x , y i r = y x + · · · + y n x n for x = ( x i ) ni =1 , y = ( y i ) ni =1 ∈ H r ,and h x , y i l = x y + · · · + x n y n for x , y ∈ H l that satisfy the inner productrelations, namely, conjugate symmetry, linearity, and positive-definiteness butwith specialties h x α + y β, z i = h x , z i α + h y , z i β when x , y , z ∈ H r h α x + β y , z i = α h x , z i + β h y , z i when x , y , z ∈ H l , for any α, β ∈ H . A set of vectors from H r and H l can be orthonormalize inparticular by the GramSchmidt process with corresponding projection operatorsproj u ( v ) := u h u , v i r h u , u i r , proj u ( v ) := h u , v i l h u , u i l u for H r and H l , respectively. Due to the above, the following definition makessense. Definition 2.4
Suppose U ∈ M ( n, H ) and U ∗ U = UU ∗ = I , then the matrix U is called unitary. Clear, that columns of U form a system of normalized vectors in H r , rows of U ∗ is a system of normalized vectors in H l .Due to the noncommutativity of quaternions, there are two types of eigenval-ues. A quaternion λ is said to be a left eigenvalue of A ∈ M ( n, H ) if A · x = λ · x ,and a right eigenvalue if A · x = x · λ for some nonzero quaternion column-vector x . Then, the set { λ ∈ H | Ax = λ x , x = ∈ H n } is called the left spectrumof A , denoted by σ l ( A ). The right spectrum is similarly defined by putting, σ r ( A ) := { λ ∈ H | Ax = x λ, x = ∈ H n } .The theory on the left eigenvalues of quaternion matrices has been investi-gated in particular in [29–31]. The theory on the right eigenvalues of quaternionmatrices is more developed [32–37]. We consider this is a natural consequenceof the fact that quaternion column vectors form a right vector space for whichleft eigenvalues seem to be ”exotic” because of their multiplying from the left.We present the some known results from the theory of right eigenvalues.It’s well known that if λ is a nonreal eigenvalue of A , so is any element in theequivalence class containing [ λ ], i.e. [ λ ] = { x | x = u − λu, u ∈ H , k u k = 1 } . Theorem 2.8 [32] Any quaternion matrix A ∈ M ( n, H ) has exactly n eigen-values which are complex numbers with nonnegative imaginary parts. h + k i , . . . , h n + k n i , where k t ≥ h t , k t ∈ R for all t = 1 , . . . , n , are said to be the standard eigenvalues of A . Theorem 2.9 [32] Let A ∈ M ( n, H ) . Then there exists a unitary matrix U such that U ∗ AU is an upper triangular matrix with diagonal entries h + k i , . . . , h n + k n i which are the standard eigenvalues of A . Corollary 2.1 [36] Let A ∈ M ( n, H ) with the standard eigenvalues h + k i , . . . , h n + k n i . Then σ r = [ h + k i ] ∪ · · · ∪ [ h n + k n i ] . Corollary 2.2 [36] Let A ∈ M ( n, H ) be given. Then, A is Hermitian if andonly if there are a unitary matrix U ∈ M ( n, H ) and a real diagonal matrix D = diag ( λ , λ , . . . , λ n ) such that A = UDU ∗ , where λ , ..., λ n are righteigenvalues of A . The right and left eigenvalues are in general unrelated [38], but it is not forHermitian matrices. Suppose A ∈ M ( n, H ) is Hermitian and λ ∈ R is its righteigenvalue, then A · x = x · λ = λ · x . This means that all right eigenvaluesof a Hermitian matrix are its left eigenvalues as well. For real left eigenvalues, λ ∈ R , the matrix λ I − A is Hermitian. Definition 2.5 If t ∈ R , then for a Hermitian matrix A the polynomial p A ( t ) =det ( t I − A ) is said to be the characteristic polynomial of A . Lemma 2.1 [15] If A ∈ M ( n, H ) is Hermitian, then p A ( t ) = t n − d t n − + d t n − − . . . + ( − n d n , where d k is the sum of principle minors of A of order k , ≤ k < n , and d n = det A . The roots of the characteristic polynomial of a Hermitian matrix are its realleft eigenvalues, which are its right eigenvalues as well.
Within the framework of the theory of column-row determinants, we have thefollowing theorem on determinantal representations of the quaternion Moore-Penrose inverse.
Theorem 2.10 [5] If A ∈ H m × nr , then the Moore-Penrose inverse A † = (cid:16) a † ij (cid:17) ∈ H n × m possess the following determinantal representations:(i) If r < min { m, n } , then a † ij = P β ∈ J r, n { i } cdet i (cid:0) ( A ∗ A ) . i (cid:0) a ∗ .j (cid:1)(cid:1) ββ P β ∈ J r, n det( A ∗ A ) ββ , (8)8 r a † ij = P α ∈ I r,m { j } rdet j (( AA ∗ ) j . ( a ∗ i. )) αα P α ∈ I r, m det( AA ∗ ) αα . (9) (ii) If r = n , then a † ij = cdet i ( A ∗ A ) . i (cid:0) a ∗ .j (cid:1) det( A ∗ A ) (10) or (9) when n < m .(iii) If r = m , then a † ij = rdet j ( AA ∗ ) j . ( a ∗ i. )det( AA ∗ ) (11) or (8) when m < n . Even though the eigenvalues of A ♯ A and AA ♯ are real and nonnegative, theyare not Hermitian in general. Therefor, the following two cases are considered,when A ♯ A and AA ♯ both or one of them are Hermitian, and when they arenon-Hermitian. Denote the ( ij )th entry of A † M,N by a ‡ ij for all i = 1 , . . . , n and j = 1 , . . . , m . Theorem 2.11 [8] Let A ∈ H m × nr . If A ♯ A or AA ♯ are Hermitian, then theweighted Moore-Penrose inverse A † M,N = (cid:16) a ‡ ij (cid:17) ∈ H n × m possess the followingdeterminantal representations, respectively,(i) If r < min { m, n } , then a ‡ ij = P β ∈ J r, n { i } cdet i (cid:16)(cid:0) A ♯ A (cid:1) . i (cid:16) a ♯.j (cid:17)(cid:17) ββ P β ∈ J r, n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) , (12) or a ‡ ij = P α ∈ I r,m { j } rdet j (cid:16) ( AA ♯ ) j . ( a ♯i. ) (cid:17) αα P α ∈ I r, m | ( AA ♯ ) αα | . (13) (ii) If rank A = n < m , then a ‡ ij = cdet i ( A ♯ A ) . i (cid:16) a ♯. j (cid:17) det( A ♯ A ) , (14) or the determinantal representation (12) can be applicable as well. iii) If rank A = m < n , then a ‡ ij = rdet j ( AA ♯ ) j. (cid:16) a ♯i. (cid:17) det( AA ♯ ) . (15) or the determinantal representation (13) can be applicable as well. Denote M = (cid:16) m ( ) ij (cid:17) , N − = (cid:16) n ( − ) ij (cid:17) , and e A := M AN − = ( e a ij ) ∈ H m × n ,then N − A ∗ M = e A ∗ = (cid:0)e a ∗ ij (cid:1) , (cid:16) M AN − (cid:17) † = e A † = (cid:16)e a † ij (cid:17) . Theorem 2.12 [8] Let A ∈ H m × nr .(i) If A ♯ A is non-Hermitian, then the weighted Moore-Penrose inverse A † M,N = (cid:16) a ‡ ij (cid:17) ∈ H n × m possess the determinantal representations(a) if r < n a ‡ ij = P k n ( − ) ik P β ∈ J r, n { i } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k ( b a .j ) (cid:17) ββ P β ∈ J r, n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) , (16) where b a .j is the j th column of N − A ∗ M ;(b) if r = n a ‡ ij = cdet i ( A ∗ MA ) .i ( b a .j )det( A ∗ MA ) , (17) where b a .j is the j th column of A ∗ M for all j = 1 , . . . , m .(ii) If AA ♯ is non-Hermitian , then A † M,N = (cid:16) a ‡ ij (cid:17) possess the determinantalrepresentation(a) if r < m , a ‡ ij = P l P α ∈ I r, m { l } rdet l (cid:16)(cid:16) e A e A ∗ (cid:17) l. ( b a i. ) (cid:17) αα · m ( ) lj P α ∈ I r, m (cid:12)(cid:12)(cid:12)(cid:16) e A e A ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) , (18) where b a i. is the i th row of N − A ∗ M ;(b) if r = m , a ‡ ij = rdet j ( AN − A ∗ ) j. ( b a i. )det( AN − A ∗ ) . (19) where b a i. is the i th row of N − A ∗ for all i = 1 , . . . , n . Cramer’s Rule for Two-sided RestrictedQuaternionic Matrix Equation
Definition 3.1
For an arbitrary matrix over the quaternion skew field, A ∈ H m × n , we denote by • R r ( A ) = { y ∈ H m × : y = Ax , x ∈ H n × } , the column right space of A , • N r ( A ) = { x ∈ H n × : Ax = 0 } , the right null space of A , • R l ( A ) = { y ∈ H × n : y = xA , x ∈ H × m } , the row left space of A , • N l ( A ) = { x ∈ H × m : xA = 0 } , the left null space of A . It is easy to see, if A ∈ H n × nn , then R r ⊕ N r = H n × , and R l ⊕ N l = H × n . Suppose that A ∈ H m × n , B ∈ H p × q . Denote R r ( A , B ) := N r ( Y ) = { Y = AXB : X n × q } , N r ( A , B ) := R r ( X ) = { X n × p : AXB = } , R l ( A , A :) = R l ( Y ) = { Y = AXB : X n × q } , N l ( A , B ) := N l ( X ) = { X n × p : AXB = } . Lemma 3.1 [17] Suppose that A ∈ H m × nr , B ∈ H p × qr , M , N , P , and Q are Hermitian positive definite matrices of order m , n , p , and q , respectively.Denote A ♯ = N − A ∗ M and B ♯ = Q − B ∗ P . If D ⊂ R r (cid:0) AA ♯ , B ♯ B (cid:1) and D ⊂ R l (cid:0) A ♯ A , BB ♯ (cid:1) , AXB = D , (20) R r ( X ) ⊂ N − R r ( A ∗ ) , N r ( X ) ⊃ P − N r ( B ∗ ) , (21) R l ( X ) ⊂ R l ( A ∗ ) M , N l ( X ) ⊃ N l ( B ∗ ) Q (22) then the unique solution of (20) with the restrictions (21)-(22) is X = A † M,N DB † P,Q . (23)In this chapter, we get determinantal representations of (23) that are intrin-sically analogs of the classical Cramer’s rule. We will consider several casesdepending on whether the matrices A ♯ A and BB ♯ are Hermitian or not. ♯ A and BB ♯ . Denote e D = A ♯ DB ♯ . Theorem 3.1
Let A ♯ A and BB ♯ be Hermitian. Then the solution (23) possessthe following determinantal representations. i) If rank A = r < n and rank B = r < p , then x ij = P β ∈ J r , n { i } cdet i (cid:0)(cid:0) A ♯ A (cid:1) . i (cid:0) d B . j (cid:1)(cid:1) ββ P β ∈ J r ,n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p (cid:12)(cid:12) ( BB ♯ ) αα (cid:12)(cid:12) , (24) or x ij = P α ∈ I r ,p { j } rdet j (cid:16)(cid:0) BB ♯ (cid:1) j . (cid:0) d A i . (cid:1)(cid:17) αα P β ∈ J r ,n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p (cid:12)(cid:12) ( BB ♯ ) αα (cid:12)(cid:12) , (25) where d B . j = X α ∈ I r ,p { j } rdet j (cid:16)(cid:0) BB ♯ (cid:1) j. (cid:16) ˜ d k. (cid:17)(cid:17) αα ∈ H n × (26) d A i . = X β ∈ J r ,n { i } cdet i (cid:16)(cid:0) A ♯ A (cid:1) .i (cid:16) ˜ d .l (cid:17)(cid:17) ββ ∈ H × p (27) are the column-vector and the row-vector, respectively. ˜ d k. and ˜ d .l are the k th row and the l th column of e D for all k = 1 , ..., n , l = 1 , ..., p .(ii) If rank A = n and rank B = p , then x i j = cdet i ( A ♯ A ) . i (cid:0) d B .j (cid:1) det( A ♯ A ) · det( BB ♯ ) , (28) or x i j = rdet j ( BB ♯ ) j. (cid:0) d A i . (cid:1) det( A ♯ A ) · det( BB ♯ ) , (29) where d B .j := (cid:16) rdet j ( BB ♯ ) j. (cid:16) ˜ d k . (cid:17)(cid:17) ∈ H n × , (30) d A i . := (cid:16) cdet i ( A ♯ A ) . i (cid:16) ˜ d .l (cid:17)(cid:17) ∈ H × p . (31) (iii) If rank A = n and rank B = r < p , then x ij = cdet i (cid:0)(cid:0) A ♯ A (cid:1) . i (cid:0) d B . j (cid:1)(cid:1) det( A ♯ A ) · P α ∈ I r ,p (cid:12)(cid:12) ( BB ♯ ) αα (cid:12)(cid:12) , (32) or x ij = P α ∈ I r ,p { j } rdet j (cid:16)(cid:0) BB ♯ (cid:1) j . (cid:0) d A i . (cid:1)(cid:17) αα det( A ♯ A ) · P α ∈ I r ,p (cid:12)(cid:12) ( BB ♯ ) αα (cid:12)(cid:12) , (33) where d B . j is (26) and d A i . is (31). iv) If rank A = r < n and rank B = p , then x i j = rdet j ( BB ♯ ) j. (cid:0) d A i . (cid:1)P β ∈ J r ,n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) · det( BB ♯ ) , (34) or x i j = P β ∈ J r , n { i } cdet i (cid:0)(cid:0) A ♯ A (cid:1) . i (cid:0) d B . j (cid:1)(cid:1) ββ P β ∈ J r ,n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) · det( BB ♯ ) , (35) where d B . j is (30) and d A i . is (27). Proof. (i) If A ∈ H m × nr , B ∈ H p × qr and r < n , r < p , then, by Theorem 2.11,the weighted Moore-Penrose inverses A † = (cid:16) a ‡ ij (cid:17) ∈ H n × m and B † = (cid:16) b ‡ ij (cid:17) ∈ H q × p possess the following determinantal representations, respectively, a ‡ ij = P β ∈ J r , n { i } cdet i (cid:16)(cid:0) A ♯ A (cid:1) . i (cid:16) a ♯.j (cid:17)(cid:17) ββ P β ∈ J r , n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) , (36) b ‡ ij = P α ∈ I r ,p { j } rdet j (cid:16) ( BB ♯ ) j . ( b ♯i. ) (cid:17) αα P α ∈ I r ,p | ( BB ♯ ) αα | . (37)By Lemma 3.1, X = A † M,N DB † P,Q and entries of X = ( x ij ) are x ij = q X s =1 m X k =1 a ‡ ik d ks ! b ‡ sj . (38)for all i = 1 , ..., n , j = 1 , ..., p .Denote by ˆ d .s the s th column of A ♯ D =: ˆ D = ( ˆ d ij ) ∈ H n × q for all s =1 , ..., q . It follows from P k a ♯. k d ks = ˆ d . s that m X k =1 a ‡ ik d ks = m X k =1 P β ∈ J r , n { i } cdet i (cid:16)(cid:0) A ♯ A (cid:1) . i (cid:16) a ♯.k (cid:17)(cid:17) ββ P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:0) A ♯ A (cid:1) ββ (cid:12)(cid:12)(cid:12) · d ks = P β ∈ J r , n { i } m P k =1 cdet i (cid:16)(cid:0) A ♯ A (cid:1) . i (cid:16) a ♯.k (cid:17)(cid:17) ββ · d ks P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:0) A ♯ A (cid:1) ββ (cid:12)(cid:12)(cid:12) = P β ∈ J r , n { i } cdet i (cid:16)(cid:0) A ♯ A (cid:1) . i (cid:16) ˆ d . s (cid:17)(cid:17) ββ P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:0) A ♯ A (cid:1) ββ (cid:12)(cid:12)(cid:12) . (39) e s. and e . s are the unit row-vector and the unit column-vector, re-spectively, such that all their components are 0, except the s th components,which are 1. Substituting (39) and (37) in (38), we obtain x ij = q X s =1 P β ∈ J r , n { i } cdet i (cid:16)(cid:0) A ♯ A (cid:1) . i (cid:16) ˆ d . s (cid:17)(cid:17) ββ P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:0) A ♯ A (cid:1) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p { j } rdet j (cid:16) ( BB ♯ ) j . ( b ♯s. ) (cid:17) αα P α ∈ I r ,p (cid:12)(cid:12)(cid:0) BB ♯ (cid:1) αα (cid:12)(cid:12) . Since ˆ d . s = n X l =1 e . l ˆ d ls , b ♯s. = p X t =1 b ♯st e t. , q X s =1 ˆ d ls b ♯st = e d lt , then we have x ij = q P s =1 p P t =1 n P l =1 P β ∈ J r , n { i } cdet i (cid:0)(cid:0) A ♯ A (cid:1) . i ( e . l ) (cid:1) ββ ˆ d ls b ♯st P α ∈ I r ,p { j } rdet j (cid:0) ( BB ♯ ) j . ( e t. ) (cid:1) αα P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:0) A ♯ A (cid:1) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p (cid:12)(cid:12)(cid:0) BB ♯ (cid:1) αα (cid:12)(cid:12) = p P t =1 n P l =1 P β ∈ J r , n { i } cdet i (cid:0)(cid:0) A ♯ A (cid:1) . i ( e . l ) (cid:1) ββ e d lt P α ∈ I r ,p { j } rdet j (cid:0) ( BB ♯ ) j . ( e t. ) (cid:1) αα P β ∈ J r , n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p (cid:12)(cid:12) ( BB ♯ ) αα (cid:12)(cid:12) . (40)Denote by d A it := X β ∈ J r , n { i } cdet i (cid:16)(cid:16) A ♯ A (cid:17) . i (cid:16)e d .t (cid:17)(cid:17) ββ = n X l =1 X β ∈ J r ,n { i } cdet i (cid:16)(cid:16) A ♯ A (cid:17) . i ( e .l ) (cid:17) ββ e d lt the t th component of a row-vector d A i . = ( d A i , ..., d A ip ) for all t = 1 , ..., p . Sub-stituting it in (40), we have x ij = p P t =1 d A it P α ∈ I r ,p { j } rdet j (cid:0) ( BB ♯ ) j . ( e t. ) (cid:1) αα P β ∈ J r , n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p | ( BB ♯ ) αα | . Since p P t =1 d A it e t. = d A i . , then it follows (25).If we denote by d B lj := p X t =1 e d lt X α ∈ I r ,p { j } rdet j (cid:16) ( BB ♯ ) j . ( e t. ) (cid:17) αα = X α ∈ I r ,p { j } rdet j (cid:16) ( BB ♯ ) j . ( e d l. ) (cid:17) αα l th component of a column-vector d B . j = ( d B j , ..., d B jn ) T for all l = 1 , ..., n and substitute it in (40), we obtain x ij = n P l =1 P β ∈ J r , n { i } cdet i (cid:0)(cid:0) A ♯ A (cid:1) . i ( e . l ) (cid:1) ββ d B lj P β ∈ J r , n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p | ( BB ♯ ) αα | . Since n P l =1 e .l d B lj = d B . j , then it follows (24).(ii) If rank A = n and rank B = p , then by Theorem 2.11 the weightedMoore-Penrose inverses A † M,N = (cid:16) a ‡ ij (cid:17) ∈ H n × m and B † P,Q = (cid:16) b ‡ ij (cid:17) ∈ H q × p possess the following determinantal representations, respectively, a ‡ ij = cdet i ( A ♯ A ) . i (cid:16) a ♯. j (cid:17) det( A ♯ A ) (41) b ‡ ij = rdet j ( BB ♯ ) j. (cid:16) b ♯i. (cid:17) det( BB ♯ ) . (42)By their substituting in (38) and pondering ahead as in the previous case, weobtain (28) and (29).(iii) If A ∈ H m × nr , B ∈ H p × qr and r = n , r < p , then, for the weightedMoore-Penrose inverses A † M,N and B † P,Q , the determinantal representations (41)and (36) are more applicable to use, respectively. By their substituting in (38)and pondering ahead as in the previous case, we finally obtain (32) and (33) aswell.(iv) In this case for A † M,N and B † P,Q , we use the determinantal representa-tions (41) and (37), respectively. (cid:3)
Corollary 3.1
Suppose that A ∈ H m × nr , D ∈ H m × p , M , N are Hermitianpositive definite matrices of order m and n , respectively, A ♯ A is Hermitian.Denote b D = A ♯ D . If D ⊂ R r ( AA ♯ ) and D ⊂ R l ( A ♯ A ) , AX = D , (43) R r ( X ) ⊂ N − R r ( A ∗ ) , R l ( X ) ⊂ R l ( A ∗ ) M , (44) then the unique solution of (43) with the restrictions (44) is X = A † M,N D which possess the following determinantal representations.(i) If rank A = r < n , then x ij = P β ∈ J r , n { i } cdet i (cid:16)(cid:0) A ♯ A (cid:1) . i (cid:16)b d . j (cid:17)(cid:17) ββ P β ∈ J r ,n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) , here b d .j are the j th column of b D for all i = 1 , ..., n , j = 1 , ..., p .(ii) If rank A = n , then x i j = cdet i ( A ♯ A ) . i (cid:16)b d .j (cid:17) det( A ♯ A ) , Proof.
The proof follows evidently from Theorem 3.1 when B be removed, andunit matrices insert instead P , Q . Corollary 3.2
Suppose that B ∈ H p × qr , D ∈ H n × q , P , and Q are Hermi-tian positive definite matrices of order p and q , respectively, BB ♯ is Hermitian.Denote ˇ D = DB ♯ . If D ⊂ R r ( B ♯ B ) and D ⊂ R l ( BB ♯ ) , XB = D , (45) N r ( X ) ⊃ P − N r ( B ∗ ) , N l ( X ) ⊃ N l ( B ∗ ) Q , (46) then the unique solution of (45) with the restrictions (46) is X = DB † P,Q which possess the following determinantal representations.(i) If rank B = r < p , then x ij = P α ∈ I r ,q { j } rdet j (cid:16)(cid:0) BB ♯ (cid:1) j . (cid:0) ˇ d i . (cid:1)(cid:17) αα P α ∈ I r ,q (cid:12)(cid:12) ( BB ♯ ) αα (cid:12)(cid:12) , where ˇ d i. are the i th row of ˇ D for all i = 1 , ..., n , j = 1 , ..., p .(ii) If rank B = p , then x i j = rdet j (cid:0) BB ♯ (cid:1) j. (cid:0) ˇ d i. (cid:1) det ( BB ♯ ) . Proof.
The proof follows evidently from Theorem 3.1 when A be removed andunit matrices insert instead M , N . ♯ A andBB ♯ . Denote e A := M AN − = ( e a ij ) ∈ H m × n , e A ∗ = N − A ∗ M and e B := P BQ − = ( e a ij ) ∈ H p × q , e B ∗ = Q − B ∗ P . Theorem 3.2
Let A ♯ A and BB ♯ be both non-Hermitian. Then the solution(23) possess the following determinantal representations. i) If rank A = r < n and rank B = r < p , then x ij = P k n ( − ) ik P β ∈ J r , n { k } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k (cid:0) d B .j (cid:1)(cid:17) ββ P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) , (47) or x ij = P l P α ∈ I r , p { l } rdet l (cid:16)(cid:16) e B e B ∗ (cid:17) l. ( d A i. ) (cid:17) αα · m ( ) lj P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) P α ∈ I r ,p (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) , (48) where d B . j = X l X α ∈ I r , p { l } rdet l (cid:16)(cid:16) e B e B ∗ (cid:17) l. ( e d t. ) (cid:17) αα · m ( ) lj ∈ H n × (49) d A i . = X k n ( − ) ik X β ∈ J r , n { k } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k (cid:16)e d .f (cid:17)(cid:17) ββ ∈ H × p (50) are the column-vector and the row-vector, respectively. ˜ d t. and ˜ d .f arethe t th row and the f th column of e D := N − A ∗ MDQ − B ∗ P = ( e d ij ) ∈ H n × p for all t = 1 , ..., n , f = 1 , ..., p .(ii) If rank A = n and rank B = p , then x i j = cdet i ( A ∗ MA ) . i (cid:0) d B .j (cid:1) det( A ∗ MA ) · det( BQ − B ∗ ) , (51) or x i j = rdet j ( BQ − B ∗ ) j. (cid:0) d A i . (cid:1) det( A ∗ MA ) · det( BQ − B ∗ ) , (52) where d B .j := (cid:16) rdet j ( BQ − B ∗ ) j. (cid:16)e d t . (cid:17)(cid:17) ∈ H n × , (53) d A i . := (cid:16) cdet i ( A ∗ MA ) . i (cid:16)e d .f (cid:17)(cid:17) ∈ H × p , (54) e d t . and e d .f are the t th row and the f th column of e D := A ∗ MDQ − B ∗ ∈ H n × p , respectively.(iii) If rank A = n and rank B = r < p , then x ij = cdet i (cid:0) ( A ∗ MA ) . i (cid:0) d B . j (cid:1)(cid:1) det( A ∗ MA ) · P α ∈ I r ,p (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) , (55)17 r x ij = P l P α ∈ I r , p { l } rdet l (cid:16)(cid:16) e B e B ∗ (cid:17) l. ( d A i. ) (cid:17) αα · m ( ) lj det( A ∗ MA ) · P α ∈ I r ,p (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) , (56) where d B . j is (49) and d A i . is (54).(iv) If rank A = r < n and rank B = p , then x i j = rdet j ( BQ − B ∗ ) j. (cid:0) d A i . (cid:1)P β ∈ J r ,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) · det( BQ − B ∗ ) , (57) or x i j = P k n ( − ) ik P β ∈ J r , n { k } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k (cid:0) d B .j (cid:1)(cid:17) ββ P β ∈ J r ,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) · det( BQ − B ∗ ) , (58) where d B . j is (53) and d A i . is (50). Proof. (i) If A ∈ H m × nr , B ∈ H p × qr are both non-Hermitian, and r < n , r < p , then, by Theorem 2.12, the weighted Moore-Penrose inverses A † = (cid:16) a ‡ ij (cid:17) ∈ H n × m and B † = (cid:16) b ‡ ij (cid:17) ∈ H q × p posses the following determinantalrepresentations, respectively, a ‡ ij = P k n ( − ) ik P β ∈ J r , n { k } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k ( b a .j ) (cid:17) ββ P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) , (59)where b a .j is the j th column of N − A ∗ M ; b ‡ ij = P l P α ∈ I r , p { l } rdet l (cid:16)(cid:16) e B e B ∗ (cid:17) l. ( b b i. ) (cid:17) αα · m ( ) lj P α ∈ I , p (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) , (60)where b b i. is the i th row of Q − B ∗ P . By Lemma 3.1, X = A † M,N DB † P,Q andentries of X = ( x ij ) are x ij = q X s =1 m X t =1 a ‡ it d ts ! b ‡ sj . (61)for all i = 1 , ..., n , j = 1 , ..., p . 18enote by b d .s the s th column of N − A ∗ MD =: b D = ( b d ij ) ∈ H n × q for all s = 1 , ..., q . It follows from P t b a . t d ts = b d . s that m X t =1 a ‡ it d ts = m X t =1 P k n ( − ) ik P β ∈ J r , n { i } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k ( b a .t ) (cid:17) ββ P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) · d ts = P k n ( − ) ik P β ∈ J r , n { i } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k (cid:16)b d .s (cid:17)(cid:17) ββ P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) . (62)Suppose e s. and e . s are the unit row-vector and the unit column-vector, respec-tively, such that all their components are 0, except the s th components, whichare 1. Substituting (62) and (60) in (61), we obtain x ij = q X s =1 P k n ( − ) ik P β ∈ J r , n { k } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k (cid:16)b d .s (cid:17)(cid:17) ββ P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) × P l P α ∈ I r , p { l } rdet l (cid:16)(cid:16) e B e B ∗ (cid:17) l. ( b b s. ) (cid:17) αα · m ( ) lj P α ∈ I , p (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) . Since b d . s = n X l =1 e . l b d ls , b b s. = p X t =1 b b st e t. , q X s =1 b d ls b b st = e d lt , then we have x ij = p P t =1 n P f =1 P k n ( −
12 ) ik P β ∈ Jr , n { k } cdet k ( ( e A ∗ e A ) . k ( e .f )) ββ e dft P l P α ∈ Ir , p { l } rdet l ( ( e B e B ∗ ) l. ( e t. ) ) αα · m ( 12 ) lj P β ∈ Jr , n | ( e A ∗ e A ) ββ | P α ∈ Ir ,p | ( e B e B ∗ ) αα | . (63)Denote by d A it := X k n ( − ) ik X β ∈ J r , n { k } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k (cid:16)e d .t (cid:17)(cid:17) ββ = n X f =1 X k n ( − ) ik X β ∈ J r , n { k } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k ( e .f ) (cid:17) ββ e d ft t th component of the row-vector d A i . = ( d A i , ..., d A ip ) for all t = 1 , ..., p .Substituting it in (63), we have x ij = p P t =1 d A it P l P α ∈ I r , p { l } rdet l (cid:16)(cid:16) e B e B ∗ (cid:17) l. ( e t. ) (cid:17) αα · m ( ) lj P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) P α ∈ I r ,p (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) . Since p P t =1 d A it e t. = d A i . , then it follows (48).If we denote by p X t =1 e d ft X l X α ∈ I r , p { l } rdet l (cid:16)(cid:16) e B e B ∗ (cid:17) l. ( e t. ) (cid:17) αα · m ( ) lj = X l X α ∈ I r , p { l } rdet l (cid:16)(cid:16) e B e B ∗ (cid:17) l. ( e d f. ) (cid:17) αα · m ( ) lj =: d B fj the f th component of the column-vector d B . j = ( d B j , ..., d B jn ) T for all f = 1 , ..., n and substitute it in (63), then x ij = n P f =1 P k n ( − ) ik P β ∈ J r , n { k } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k ( e .f ) (cid:17) ββ d B fj P β ∈ J r , n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p | ( BB ♯ ) αα | . Since n P f =1 e .f d B fj = d B . j , then it follows (47).(ii) If rank A = n and rank B = p , then by Theorem 2.12 the weightedMoore-Penrose inverses A † M,N = (cid:16) a ‡ ij (cid:17) ∈ H n × m and B † P,Q = (cid:16) b ‡ ij (cid:17) ∈ H q × p possess the following determinantal representations, respectively, a ‡ ij = cdet i ( A ∗ MA ) .i ( b a .j )det( A ∗ MA ) , (64) b ‡ ij = rdet j ( BQ − B ∗ ) j. ( b b i. )det( BQ − B ∗ ) . (65)where b a .j is the j th column of A ∗ M for all j = 1 , . . . , m , and b b i. is the i th rowof Q − B ∗ for all i = 1 , . . . , n .By their substituting in (61), we obtain x ij = p P t =1 n P f =1 cdet i ( A ∗ MA ) .i ( e .f ) e d ft rdet j ( BQ − B ∗ ) j. ( e t. )det ( A ∗ MA )det( BQ − B ∗ ) , e d ft is the ( f t )th entry of e D := A ∗ MDQ − B ∗ in this case. Denote by d A it := cdet i ( A ∗ MA ) .i ( e d .t )the t th component of the row-vector d A i . = ( d A i , ..., d A ip ) for all t = 1 , ..., p .Substituting it in (63), it follows (51).Similarly, we can obtain (52).(iii) If A ∈ H m × nr , B ∈ H p × qr and r = n , r < p , then, for the weightedMoore-Penrose inverses A † M,N and B † P,Q , the determinantal representations (64)and (59) are more applicable to use, respectively. By their substituting in (61)and pondering ahead as in the previous case, we finally obtain (55) and (56) aswell.(iv) In this case for A † M,N and B † P,Q , we use the determinantal representa-tions (59) and (65), respectively. (cid:3)
Corollary 3.3
Suppose that A ∈ H m × nr , D ∈ H m × p , M , N are Hermitianpositive definite matrices of order m and n , respectively, and A ♯ A is non-Hermitian. If D ⊂ R r ( AA ♯ ) and D ⊂ R l ( A ♯ A ) , then the unique solution X = A † M,N D of the equation AX = D with the restrictions (44) possess thefollowing determinantal representations.(i) If rank A = r < n , then x ij = P k n ( − ) ik P β ∈ J r , n { i } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k (cid:16)e d .j (cid:17)(cid:17) ββ P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) , where e d .j are the j th column of e D = N − A ∗ MD for all i = 1 , ..., n , j = 1 , ..., p .(ii) If rank A = n , then x i j = cdet i ( A ∗ MA ) . i (cid:16)e d .j (cid:17) det ( A ∗ MA ) , where e d .j are the j th column of e D = A ∗ MD . Proof.
The proof follows evidently from Theorem 3.2 when B be removed andunit matrices insert instead P , Q . Corollary 3.4
Suppose that B ∈ H p × qr , D ∈ H n × q , P , and Q are Hermi-tian positive definite matrices of order p and q , respectively, and BB ♯ is non-Hermitian. If D ⊂ R r ( B ♯ B ) and D ⊂ R l ( BB ♯ ) , then the unique solution X = DB † P,Q of the equation XB = D with the restrictions (46) possess thefollowing determinantal representations. i) If rank B = r < p , then x ij = P α ∈ I r ,q { j } rdet j (cid:18)(cid:16) e B e B ∗ (cid:17) j . (cid:16)e d i . (cid:17)(cid:19) αα P α ∈ I r ,q (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) , where e d i. are the i th row of e D = DQ − B ∗ P for all i = 1 , ..., n , j =1 , ..., p .(ii) If rank B = p , then x i j = rdet j (cid:0) BQ − B ∗ (cid:1) j. (cid:16)e d i. (cid:17) det ( BQ − B ∗ ) , (66) where e d i. are the i th row of e D = DQ − B ∗ . Proof.
The proof follows evidently from Theorem 3.2 when A be removed andunit matrices insert instead M , N . In this subsection we consider mixed cases when only one from the pair A ♯ A and BB ♯ is non-Hermitian. We give this theorems without proofs, since theirproofs are similar to the proof of Theorems 3.1 and 3.2. Theorem 3.3
Let A ♯ A be Hermitian and BB ♯ be non-Hermitian. Then thesolution (23) possess the following determinantal representations.(i) If rank A = r < n and rank B = r < p , then x ij = P β ∈ J r , n { i } cdet i (cid:0)(cid:0) A ♯ A (cid:1) . i (cid:0) d B . j (cid:1)(cid:1) ββ P β ∈ J r ,n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) , or x ij = P l P α ∈ I r , p { l } rdet l (cid:16)(cid:16) e B e B ∗ (cid:17) l. ( d A i. ) (cid:17) αα · m ( ) lj P β ∈ J r ,n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) , where d B . j = X l X α ∈ I r , p { l } rdet l (cid:16)(cid:16) e B e B ∗ (cid:17) l. ( e d t. ) (cid:17) αα · m ( ) lj ∈ H n × (67)22 A i . = X β ∈ J r ,n { i } cdet i (cid:16)(cid:0) A ♯ A (cid:1) .i (cid:16)e d .f (cid:17)(cid:17) ββ ∈ H × p (68) are the column-vector and the row-vector, respectively. ˜ d t. and e d .f are the t th row and the f th column of e D := A ♯ DQ − B ∗ P for all t = 1 , ..., n , f = 1 , ..., p .(ii) If rank A = n and rank B = p , then x i j = cdet i ( A ♯ A ) . i (cid:0) d B .j (cid:1) det( A ♯ A ) · det( BQ − B ∗ ) , or x i j = rdet j ( BQ − B ∗ ) j. (cid:0) d A i . (cid:1) det( A ♯ A ) · det( BQ − B ∗ ) , where d B .j := (cid:16) rdet j ( BQ − B ∗ ) j. (cid:16) ˜ d t . (cid:17)(cid:17) ∈ H n × , (69) d A i . := (cid:16) cdet i ( A ♯ A ) . i (cid:16) ˜ d .f (cid:17)(cid:17) ∈ H × p , (70)˜ d t . , ˜ d .f are the t th row and f th column of ˜ D = A ♯ DQ − B ∗ .(iii) If rank A = n and rank B = r < p , then x ij = cdet i (cid:0)(cid:0) A ♯ A (cid:1) . i (cid:0) d B . j (cid:1)(cid:1) det( A ♯ A ) · P α ∈ I r ,p (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) , or x ij = P l P α ∈ I r , p { l } rdet l (cid:16)(cid:16) e B e B ∗ (cid:17) l. ( d A i. ) (cid:17) αα · m ( ) lj det( A ♯ A ) · P α ∈ I r ,p (cid:12)(cid:12)(cid:12)(cid:16) e B e B ∗ (cid:17) αα (cid:12)(cid:12)(cid:12) , where d B . j is (67) and d A i . is (70).(iv) If rank A = r < n and rank B = p , then x i j = rdet j ( BQ − B ∗ ) j. (cid:0) d A i . (cid:1)P β ∈ J r ,n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) · det( BQ − B ∗ ) , or x i j = P β ∈ J r , n { i } cdet i (cid:0)(cid:0) A ♯ A (cid:1) . i (cid:0) d B . j (cid:1)(cid:1) ββ P β ∈ J r ,n (cid:12)(cid:12)(cid:12) ( A ♯ A ) ββ (cid:12)(cid:12)(cid:12) · det( BQ − B ∗ ) , where d B . j is (69) and d A i . is (68). heorem 3.4 Let A ♯ A be non-Hermitian, and BB ♯ be Hermitian. Denote e D := e A ∗ DB ♯ . Then the solution (23) possess the following determinantal rep-resentations.(i) If rank A = r < n and rank B = r < p , then x ij = P k n ( − ) ik P β ∈ J r , n { k } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k (cid:0) d B .j (cid:1)(cid:17) ββ P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12) P α ∈ I r ,p (cid:12)(cid:12) ( BB ♯ ) αα (cid:12)(cid:12) , or x ij = P α ∈ I r ,p { j } rdet j (cid:16)(cid:0) BB ♯ (cid:1) j . (cid:0) d A i . (cid:1)(cid:17) αα P β ∈ J r , n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) P α ∈ I r ,p (cid:12)(cid:12) ( BB ♯ ) αα (cid:12)(cid:12) , where d B . j = X α ∈ I r ,p { j } rdet j (cid:16)(cid:0) BB ♯ (cid:1) j . (cid:16)e d t . (cid:17)(cid:17) αα ∈ H n × (71) d A i . = X k n ( − ) ik X β ∈ J r , n { k } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k (cid:16)e d .f (cid:17)(cid:17) ββ ∈ H × p (72) are the column-vector and the row-vector, respectively. ˜ d t. and ˜ d .f are the t th row and the f th column of e D := N − A ∗ MDB ♯ for all t = 1 , ..., n , f = 1 , ..., p .(ii) If rank A = n and rank B = p , then x i j = cdet i ( A ∗ MA ) . i (cid:0) d B .j (cid:1) det( A ∗ MA ) · det ( BB ♯ ) , or x i j = rdet j (cid:0) BB ♯ (cid:1) j. (cid:0) d A i . (cid:1) det( A ∗ MA ) · det ( BB ♯ ) , where d B .j := (cid:16) rdet j (cid:0) BB ♯ (cid:1) j. (cid:16) ˜ d t . (cid:17)(cid:17) ∈ H n × , (73) d A i . := (cid:16) cdet i ( A ∗ MA ) . i (cid:16) ˜ d .f (cid:17)(cid:17) ∈ H × p , (74)˜ d t . , ˜ d .f are the t th row and f th column of ˜ D = A ∗ MDB ♯ . iii) If rank A = n and rank B = r < p , then x ij = cdet i (cid:0) ( A ∗ MA ) . i (cid:0) d B . j (cid:1)(cid:1) det( A ∗ MA ) · P α ∈ I r ,p (cid:12)(cid:12) ( BB ♯ ) αα (cid:12)(cid:12) , or x ij = P α ∈ I r ,p { j } rdet j (cid:16)(cid:0) BB ♯ (cid:1) j . (cid:0) d A i . (cid:1)(cid:17) αα det( A ∗ MA ) · P α ∈ I r ,p (cid:12)(cid:12) ( BB ♯ ) αα (cid:12)(cid:12) , where d B . j is (71) and d A i . is (74).(iv) If rank A = r < n and rank B = p , then x i j = rdet j (cid:0) BB ♯ (cid:1) j. (cid:0) d A i . (cid:1)P β ∈ J r ,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) · det ( BB ♯ ) , or x i j = P β ∈ J r , n { i } cdet i (cid:0)(cid:0) A ♯ A (cid:1) . i (cid:0) d B . j (cid:1)(cid:1) ββ P β ∈ J r ,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) · det( BQ − B ∗ ) , where d B . j is (73) and d A i . is (72). Let us consider the restricted matrix equation
AXB = D , R r ( X ) ⊂ N − R r ( A ∗ ) , N r ( X ) ⊃ P − N r ( B ∗ ) , R l ( X ) ⊂ R l ( A ∗ ) M , N l ( X ) ⊃ N l ( B ∗ ) Q (75)where A = (cid:18) k − j j k (cid:19) , N = − j j , M = (cid:18) k − k (cid:19) , D = (cid:18) i − j k − k j (cid:19) , B = (cid:18) k − j j i (cid:19) , Q = i − i − j j , P = (cid:18) . − . j . j . (cid:19) . Since A ∗ = − k i − j − k , B ∗ = − k j − j − i , AA ∗ ) = det (cid:18) − i + ki − k (cid:19) = 4 , det( BB ∗ ) = det (cid:18) − j + kj − k (cid:19) = 4 , then rank A = rank B = 2.Due to Theorem 2.4, we can be obtain the inverses Q − = 13 − i − k i j k − j , N − = 19 j − j . It is easy to verify that the both matrices A ♯ A = N − A ∗ MA and BB ♯ = BQ − B ∗ P are not Hermitian. Hence, we shall find the solution of (75) by (57).So, x i j = rdet j ( BQ − B ∗ ) j. (cid:0) d A i . (cid:1)P β ∈ J , (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) · det( BQ − B ∗ ) , i = 1 , , , j = 1 , , (76)where d A i . is (50), namely d A i . = X k n ( − ) ik X β ∈ J , { k } cdet k (cid:16)(cid:16) e A ∗ e A (cid:17) . k (cid:16)e d .f (cid:17)(cid:17) ββ ∈ H × . (77)To obtain N − , we firstly find the eigenvalues of N which are the roots of thecharacteristic polynomial p ( λ ) = det λ − j λ − − j λ − = λ − λ + 49 λ − ⇒ λ = 1 ,λ = 4 ,λ = 9 . By computing the associated eigenvectors and after their orthonormalization,we obtain the unitary matrix U whose columns are this eigenvectors. U = . − . j . . j . . j . − . j . Finally, we have N = U ∗ DU = − j j , where D = diag(1 , ,
3) is the diagonal matrix with 1 , , N − = j − j . M = (cid:18) k − k (cid:19) , P = (cid:18) − jj (cid:19) . Further, we find e D := N − A ∗ MDQ − B ∗ P = − + i + j − k − i + j − k + i + j − k + i − j − k + i − j + k − + i − j − k , e A = M AN − = (cid:18) + j + 2 k − i − − i + k i − k j i + j + k (cid:19) , e A ∗ A = − i + 3 j − k i + j + 3 k i − j + k
54 32 − i + k − i − j − k + i − k . X β ∈ J , (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) = det (cid:18) − i + 3 j − k i − j + k (cid:19) +det (cid:18)
54 32 − i + k + i − k (cid:19) + det (cid:18) i + j + 3 k − i − j − k (cid:19) =54 + 54 + 2 = 4 . , det (cid:0) BQ − B ∗ (cid:1) = det (cid:18)
10 1 − i − j + k i + 4 j − k (cid:19) = 8 . Now, we obtain components of the row-vectors (77), d A . = (cid:0) d A , d A (cid:1) , d A = 23 (cid:18) cdet (cid:18) − + i + j − k − i + 3 j − k + i + j − k (cid:19) +cdet (cid:18) − + i + j − k i + j + 3 k + i − j + k (cid:19)(cid:19) − j (cid:18) cdet (cid:18) − + i + j − k − i − j − k + i − j + k (cid:19) +cdet (cid:18)
54 152 + i + j − k + i − k + i − j + k (cid:19)(cid:19) = − − i + 136 j − k. Similarly, we obtain d A = − − i + j + k . Moreover, d A . = (cid:18) i + 109318 j + 179512 k, − i − j − k (cid:19) d A . = (cid:18) − i + 2536 j + 2999 k, − i − j + 24512 k (cid:19) . Finally, we have 27 = rdet ( BQ − B ∗ ) . (cid:0) d A . (cid:1)P β ∈ J , (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e A ∗ e A (cid:17) ββ (cid:12)(cid:12)(cid:12)(cid:12) · det( BQ − B ∗ ) =136 rdet (cid:18) − − i + j − k − − i + j + k i + 4 j − k (cid:19) = − i − j + 173144 k. Similarly, we obtain x = 19288 − i − j + 3247864 k, and x = 1162324 − i − j + 1759432 k, x = 1631432 + 1285324 i − j − k,x = 127864 + 83864 i − j − k, x = 3111296 + 367162 i − j + 7736 k. Note that we used Maple with the package CLIFFORD in the calculations.
In this paper, previously obtained determinantal representations of the quater-nion weighted Moore-Penrose inverse have been used to derive explicit deter-minantal representation formulas for the solution of the two-sided restrictedquaternionic matrix equation,
AXB = D , within the framework of the theoryof column-row determinants (also previously introduced by the author). References [1] R. A. Penrose, Generalized inverse for matrices, Proc. Camb. Philos. Soc. (1955) 406–413.[2] A. Ben-Israel, T.N.E. Grenville, Generalized Inverses: Theory and Appli-cations. Springer- Verlag, Berlin, 2002.[3] K.M. Prasad, R.B. Bapat, A note of the Khatri inverse, Sankhya: IndianJ. Stat. (1992) 291-295.[4] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. (1997) 21-57.[5] I.I. Kyrchei, Determinantal representation of the Moore-Penrose inversematrix over the quaternion skew field, J. Math. Sci. (1) (2012) 23-33.286] C.F. Van Loan, Generalizing the singular value decomposition, SIAM J.Numer. Anal. (1976) 76–83.[7] E. F. Galba, Weighted singular decomposition and weighted pseudoinver-sion of matrices, Ukr. Math. J. (10) (1996) 1618-1622.[8] I.I. Kyrchei, Weighted singular value decomposition and determinantalrepresentations of the quaternion weighted Moore-Penrose inverse, Appl.Math. Comput. (2017) 1-16.[9] P. Stanimirovic’, M. Stankovic’, Determinantal representation of weightedMoore-Penrose inverse, Mat. Vesnik (1994) 41-50.[10] X. Liu, Y. Yu, H. Wang, Determinantal representation of weighted gener-alized inverses, Appl. Math. Comput. (7) (2011) 3110-3121.[11] X. Liu, G. Zhu, G. Zhou, Y. Yu, An analog of the adjugate matrix forthe outer inverse A (2) T,S , Math. Problem. in Eng. , Article ID 591256(2012) 14 pages.[12] I.I. Kyrchei, Analogs of the adjoint matrix for generalized inverses andcorresponding Cramer rules, Linear Multilinear Algebra (4) (2008) 453-469.[13] I.I. Kyrchei, Cramer’s rule for generalized inverse solutions, In: Advancesin Linear Algebra Research, I.I. Kyrchei (Ed.), Nova Sci. Publ., New York,pp. 79-132, 2015.[14] I.I. Kyrchei, Cramer’s rule for quaternion systems of linear equations. Fun-damentalnaya i Prikladnaya Matematika (4) (2007) 67-94.[15] I.I. Kyrchei, The theory of the column and row determinants in a quater-nion linear algebra, In: Advances in Mathematics Research , A.R.Baswell (Ed.), Nova Sci. Publ., New York, pp. 301-359, 2012.[16] G. Song, Q. Wang, H. Chang, Cramer rule for the unique solution of re-stricted matrix equations over the quaternion skew field, Comput Math.Appl. (2011) 1576-1589.[17] G.J. Song, Determinantal representation of the generalized inverses over thequaternion skew field with applications, Appl. Math. Comput. (2012)656–667.[18] I.I. Kyrchei, Explicit representation formulas for the minimum norm leastsquares solutions of some quaternion matrix equations, Linear AlgebraAppl. (1) (2013) 136–152.[19] I.I. Kyrchei, Determinantal representations of the Drazin inverse over thequaternion skew field with applications to some matrix equations, Appl.Math. Comput. (2014) 193–207.2920] I.I. Kyrchei, Determinantal representations of the W-weighted Drazin in-verse over the quaternion skew field, Appl. Math. Comput. (2015)453–465.[21] I.I. Kyrchei, Explicit determinantal representation formulas of W-weightedDrazin inverse solutions of some matrix equations over the quaternion skewfield, Math. Problem. in Eng. Article ID 8673809 (2016) 13 pages.[22] I.I. Kyrchei, Determinantal representations of the Drazin and W-weightedDrazin inverses over the quaternion skew field with applications, In:Quaternions: Theory and Applications, S. Griffin (Ed.), New York: NovaSci. Publ., pp.201-275, 2017.[23] A. Kleyn, I. Kyrchei, Relation of row-column determinants with quaside-terminants of matrices over a quaternion algebra, In: Advances in LinearAlgebra Research,I. Kyrchei (Ed.), Nova Sci. Publ., New York, pp. 299-324,2015.[24] G.J. Song, C.Z. Dong, New results on condensed Cramers rule for thegeneral solution to some restricted quaternion matrix equations, J. Appl.Math. Comput. (2017) 321–341.[25] G.J. Song, Bott-Duffin inverse over the quaternion skew field with applica-tions, J. Appl. Math. Comput. (2013) 377-392.[26] G.J. Song, Characterization of the W-weighted Drazin inverse over thequaternion skew field with applications, Electron. J. Linear Algebra (2013) 1–14.[27] Y. Wei, H. Wu, The representation and approximation for the weightedMoore-Penrose inverse, Appl. Math. Comput. (2001) 17-28.[28] I.V. Sergienko, E.F. Galba, V.S. Deineka,Limiting representations ofweighted pseudoinverse matrices with positive definite weights. Problemregularization, Cybernetics and Systems Analysis (6) (2003) 816-830.[29] L. Huang, W. So, On left eigenvalues of a quaternionic matrix, LinearAlgebra Appl. (2001) 105-116.[30] W. So, Quaternionic left eigenvalue problem, Southeast Asian Bulletin ofMathematics (2005) 555-565.[31] R. M. W. Wood, Quaternionic eigenvalues, Bull. Lond. Math. Soc. (1985) 137-138.[32] J.L. Brenner, Matrices of quaternions, Pac. J. Math. (1951) 329-335.[33] E. Mac´ıas-Virg´os, M.J. Pereira-S´aez, A topological approach to left eigen-values of quaternionic matrices, Linear Multilinear Algebra (2) (2014)139–158. 3034] A. Baker, Right eigenvalues for quaternionic matrices: a topological ap-proach, Linear Algebra Appl. (1999) 303-309.[35] T. Dray, C. A. Manogue, The octonionic eigenvalue problem, Advances inApplied Clifford Algebras (2) (1998) 341-364.[36] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. (1997) 21-57.[37] D.R. Farenick, B.A.F. Pidkowich, The spectral theorem in quaternions,Linear Algebra Appl. (2003) 75-102.[38] F. O. Farid, Q.W. Wang, F. Zhang, On the eigenvalues of quaternion ma-trices, Linear Multilinear Algebra59