Research Article Eigenvector-Free Solutions to the Matrix Equation AXB H =E with Two Special Constraints

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1 Applied Mathematics Volume 03 Article ID pages Research Article Eigenvector-Free Solutions to the Matrix Equation AXB =E with Two Special Constraints Yuyang Qiu College of Statistics and Mathematics Zhejiang Gongshang University angzhou 3008 China Correspondence should be addressed to Yuyang Qiu; yuyangqiu77@63.com Received March 03; Accepted 8 September 03 Academic Editor: Qing-Wen Wang Copyright 03 Yuyang Qiu. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. The matrix equation AXB =Ewith SX = XR or PX = sxq constraint is considered where S R are ermitian idempotent P Q are ermitian involutory and s=±. By the eigenvalue decompositions of S RtheequationAXB =Ewith SX = XR constraint is equivalently transformed to an unconstrained problem whose coefficient matrices contain the corresponding eigenvectors with which the constrained solutions are constructed. The involved eigenvectors are released by Moore-Penrose generalized inverses and the eigenvector-free formulas of the general solutions are presented. By choosing suitable matrices S R we also present the eigenvector-free formulas of the general solutions to the matrix equation AXB =Ewith PX = sxq constraint.. Introduction In [] Chen has denoted a square matrix X the reflexive or antireflexive matrix with respect to P by PX = XP or PX = XP () where the matrix P C n n is ermitian involutory. e also pointed out that these matrices possessed special properties and had wide applications in engineering and scientific computations [ ]. So solving the matrix equation or matrix equations with these constraints is maybe interesting [3 4]. In this paper we consider the matrix equation with constraint AXB =E () PX=sXQ or SX = XR (3) where the matrices A C m n B C p n E C m p the ermitian involutory matrices P Q C n n theermitian idempotent matrices S R C n n andthescalarss=±. Equation () with different constraints such as symmetry skew-symmetry and PX = ±XP was discussed in [9 5 ] where existence conditions and the general solutions to the constrained equation were presented. By generalized singular value decomposition (GSVD) [ 3] the authors of [5 7] simplified the matrix equation by diagonalizing the coefficient matrices and block-partitioned the new variable matrices into several block matrices then imposed the constrained condition on subblocks and determined the unknown subblocks separately for () withsymmetric constraint. A similar strategy was also used in [8]; the authors achieved symmetric skew-symmetric and positive semidefinite solutions to () by quotient singular value decomposition (QSVD) [4 5]. Moreover in [0] CCD [6] was used for establishing a formula of the general solutions to () with diagonal constraint. In [9] we have presented an eigenvector-free solution to the matrix equation () with constraint PX = ±XPwhere we represented its general solution and existence condition by g-inverses of the matrices A B and P. Note that the ginverses are always not unique and they can be generalized to the Moore-Penrose generalized inverses. Moreover the constraint which guarantees the eigenvector-free expressions canbemaybeimprovedfurther.sointhispaperwefocus on () with generalized constraint PX = sxq or another constraint SX = XR; our ideas are based on the following observations. () If we set S= (I+P) R = (I+sQ) (4)

2 Applied Mathematics then S and R are both ermitian idempotent. The above fact implies PX = sxq is the special case of SX = XR.Soweonlydiscuss() withsx = XR constraint and construct the PX=sXQconstrained solution by selecting suitable matrices R Q as (4). () With the eigenvalue decompositions (EVDs) of the ermitian matrices R S matrixx with SX = XR constraint can be rewritten in (lower dimensional) two free variables X and Y. And the corresponding constrained problem can be equivalently transformed to an unconstrained equation A X B + A Y B =E (5) with given coefficient matrices A i B i i=(one can see the details of this discussion in Section ). (3) The general solutions and existence conditions of (5) can be represented by the Moore-Penrose generalized inverses of A i B i i=[ ]. owever the formulas above are maybe not simpler because the coefficient matrices contain the eigenvectors of S R. In fact the ermitian idempotence of the matrices S R implies they only have two clusters different eigenvalues and their corresponding eigenvectors appear in the expression of general solutions and existence conditions can be easily represented by S R themselves. So we present a simple and eigenvectorfree formulation for the constrained general solution. The rest of this paper is organized as follows. In Section we give the general solutions and the existence condition to () withsx = XR constraint by the EVDs of S R. In Section 3 we present the corresponding eigenvector-free representations. Equation () withpx = sxq constraint is regarded as the special case of ()withsx = XR constraint and its eigenvector-free representation is given in Section 4. Numerical examples are given in Section 5 to display the effectiveness of our theorems. We will use the following notations in the rest of this paper. Let C m n denote the space of complex m nmatrix. For a matrix A A and A denote its transpose and Moore- Penrose generalized inverse respectively. Matrix I n is identity matrix with order n; O m n refers to m nzero matrix and O n is the zero matrix with order n.foranymatrixa C m n we also denote So P A =AA K A =I m P A. (6) P A =A A K A =I n P A. (7). Solution to () with SX=RX Constraint by the EVDs For the ermitian idempotent matrices S Rlet S=Udiag (I k O n k )U R = Vdiag (I l O n l )V (8) be their two eigenvalue decompositions with unitary matrices U Vrespectively.ThenSX = XR holds if and only if diag (I k O n k ) X = X diag (I l O n l ) (9) where X =U XV. And the constrained solution X can be expressed in X=Udiag ( X Y) V X C k l Y C (n k) (n l). (0) Partitioning U = [U U ] V = [V V ] and using the transformations (0) () with SX = XR constraint is equivalent to the following unconstrained problem: where A X B + A Y B =E () A =AU B =BV A =AU B =BV. () For the unconstrained problem () we introduce the results about its existence conditions and expression of solutions. Lemma. Given A C m n B C p q C C m r D C s q and E C m q the linear matrix equation AXB + CYD = E is consistent if and only if P G K A EP D =K A E P C EK B P J =EK B (3) or equivalently if and only if K G K A E=0 K A EK D =0 (4) K C EK B =0 EK B K J =0 where G=K A C and J=DK B.Andarepresentationofthe general solution is with Y=G K A ED +T P G TP D X=A (E CYD) B +Z P A ZP B (5) T=(CK G ) (I m CG K A )EK B J +W P (CKG ) WP J (6) where the matrices W C r s and Z C n p are arbitrary. The lemma is easy to verify; we can turn to [7] for details. The difference between them is that we replace the ginverse in the theorem of [7] bythecorrespondingmoore- Penrose generalized inverse and the expression of solutions is complicated relatively. owever compared with the multiformity of the g-inverses the Moore-Penrose generalized inverse involved representation is unique and fixed. Apply Lemma on the unconstrained problem () we have the following theorem.

3 Applied Mathematics 3 Theorem. The matrix equation AXB =Ewith constraint SX = XR is consistent if and only if the involved eigenvectors in detailed expressions. With the first equality in (8) we have where P G K A EP B =K A E P A EK B P J =EK B (7) U U =S U U =I n S V V =R V V =I n R. (3) G =K A A J = B K B. (8) In the meantime a general solution is given by Y = G K A E B +( A K G ) (I m A G K A )EK B J P G ( A K G ) (I m A G K A )EK B J P B +W P G WP B P ( A K G )WP J + P G P ( A K G )WP J P B X = A (E A Y B ) B +Z P A ZP B where the matrices W and Z are arbitrary. (9) In order to separate Y from X of the second equality in (9) we substitute Y into X. Let Y = G K A E B +( A K G ) (I m A G K A )EK B J Note that U i (AU i ) is the Moore-Penrose generalized inverse of AU i U i whichgives P A i = A i A i =(AU i U i )(AU i U i ) =A i A i = P Ai (4) where Then Set A =AU U =AS A =AU U =A(I n S). (5) K A i =I m P A i =I m P Ai =K Ai GU =K A A. (6) B =BV V = BR B =BV V =B(I n R) (7) and denote G=K A A J = B K B. (8) P G ( A K G ) (I m A G K A )EK B J P B X = A E B A A Y B B (0) It is not difficult to verify that together with V J =J GU =G (9) together with B B B =( B B ) B = B () A K G ( A K G ) A K G = A K G. P G = GU ( GU ) = P G P J =(V J) (V J) = P J. Then the first equality of (7)canberewrittenas (30) Then (9)can be rewritten as Y =Y +W P G WP B P ( A K G )WP J and the other can be rewritten as P G K A EP B =K A E (3) + P G P ( A K G )WP J P B X =X +Z P A ZP B A A K G WK J B B. () P A EK B P J =EK B. (3) Now we consider the simplification of the general solution X given by (0) which can be rewritten as 3. Eigenvector-Free Formulas of the General Solutions to () with SX=XR Constraint The existence conditions and the expression of the general solutiongiven in Theorem contain the eigenvector matrices of S R respectively. This implies that the eigenvalue decompositions will be included. In this section we intend to release Note that X=U XV +U YV. (33) U G =( GU ) =G K G U =U K G U A =A. (34)

4 4 Applied Mathematics Together with (6) U (W P G WP B P ( A K G )WP J U Y V =U ( G K A E B +( A K G ) +P G P ( A K G )WP J P B )V (I m A G K A )EK B J =U WV P G U WV P B P G ( A K G ) (I m A G K A )EK B J P B =G K A EB +(A K G ) )V (35) Letting P (A K G ) U WV P J + P G P (A K G ) U WV P JP B. (39) (I m A G K A )EK B J P G (A K G ) (I m A G K A )EK B J P B so we can represent U j Y V by a given expression of A i B i E.Let f(a A B B E)=G K A EB +(A K G ) ence we have U j Y V U X V =A EB Since =A EB (I m A G K A )EK B J P G (A K G ) (I m A G K A )EK B J P B. (36) =f(a A B B E) A A U Y V B B A A f(a A B B E)B B. (37) V K J =V (I n l P J )=(I p V J(V J) )V =K J V (38) U ZV +U WV =F (40) it is not difficult for us to verify SF = FR.Togetherwith A U =0 A U =0 V B =0 V B =0 (4) the following equality holds: Note that Then P A U ZV P B + P G U WV P B =(P A + P G )(U WV +U ZV ) (P B + P B ) =(P A + P G )F(P B + P B ). (4) GU =0 A K G U =0. (43) A K G U WV =A K G (U WV +U ZV )=A K G F. (44) ence A A K G U WV K JB B =A A K G FK J B B P (A K G ) U WV P J P G P (A K G ) U WV P JP B = P (A K G ) FP J P G P (A K G ) FP JP B. Substitutingtheexpressionsaboveinto(33)yieldsthat X=A EB +f(a A B B E) (45) then A A f(a A B B E)B B +F (P A + P G )F(P B + P B ) (46) U (Z P A ZP B A A K G WK J B =U ZV P A U ZV P B A A K G U WV K JB B B )V A A K G FK J B B P (A K G ) FP J + P G P (A K G ) FP JP B. We have the following theorem.

5 Applied Mathematics 5 Theorem 3. Let A =AS B = BR A =A(I n S) B =B(I n R). (47) The matrix equation () with constraint SX = XR is consistent if and only if with P G K A EP B =K A E P A EK B P J =EK B (48) G=K A A J = B K B. (49) In the meantime a general solution is given by X=A EB +f(a A B B E) A A f(a A B B E)B B +F (P A + P G )F(P B + P B ) A A K G FK J B B P (A K G ) FP J + P G P (A K G ) FP JP B (50) where the arbitrary matrix F satisfies SF = FR and f(a A B B E)is determined by (36). 4. Eigenvector-Free Formulas of the General Solutions to () with PX=sXQ Constraint For this constraint if we set S and R as (4) it is not difficult to verify that S R are ermitian idempotent and the constraint PX = sxq is equivalent to In the meantime a general solution is given by X=A EB +f(a A B B E) A A f(a A B B E)B B +F (P A + P G )F(P B + P B ) A A K G FK J B B P (A K G ) FP J + P G P (A K G ) FP JP B (55) where the arbitrary matrix F satisfies PF = sfq and f(a A B B E)is determined by (36). 5. Numerical Examples In this section we present some numerical examples to illustrate the effectiveness of Theorems 3 and 4. For simplicity we set m=n=pand restrict the coefficient matrices A B and the right-hand-sided matrix E to R n n.thecoefficient matrices A B are randomly constructed by A=Udiag (σ...σ n )V T (56) where the orthogonal matrices U and V are constructed as follows: [U temp] =qr ( rand (n)) [V temp] =qr ( rand (n)) (57) and the singular values {σ i } will be chosen at interval (0 ). For the computational value X of () with constraint PX = sxq or SX = XR the residual error ε X thepq-commuting error ε PQ SR-commuting error ε SR andconsistenterror Cond err are denoted by ε X = E AXB F ε PQ = PX sxq F SX = XR. (5) By Theorem 3 we have the following theorem. ε SR = SX XR F Cond err = max { P GK A EP B K A E F (58) Theorem 4. Let P A EK B P J EK B F }. A = A(I n +P) B = B(I n +sq) A = A(I n P) B = B(I n sq). (5) Example. In this example we test the solutions to () with SX = XQ constraint by Theorem 3. The coefficient matrices A B are constructed as in (56) and the right-hand-sided matrix E is constructed as follows: The matrix equation () with constraint PX = sxq is consistentifandonlyif P G K A EP B =K A E P A EK B P J =EK B (53) where X satisfies E=AX B (59) RX =X S (60) with G=K A A J = B K B. (54) and S R are symmetric idempotent. That implies that the constrained equation () is consistent so the residual error ε X and consistent error Cond err should be zero with the computational value X.

6 6 Applied Mathematics Table : Variant matrix sizes n for the solutions to ()withsx = XR constraint. n CPU (s) ε X ε SR Cond err Table : Variant matrix sizes n for solutions to () withpx = XQ constraint. n CPU (s) ε X ε PQ Cond err For different n the residual error ε X SR-commuting error ε SR and consistent errors Cond err can reach the precision 0 09 but all of them seem not to depend on the matrix size n very much and the CPU time also grows quickly as n increases. In Table welistthecputimeε X ε SR and Cond err respectively. Example. We test the solutions to () with PX = XQ constraint by Theorem 4. The test matrices A B and E are constructed as in (56)withX satisfying where X satisfies E=AX B (6) PX =X Q (6) and P Q are symmetric involutory. For different nthenumericalresultissimilarto thoseof Example ;thatistheresidualerrorε X PQ-commuting error ε PQ and consistent errors Cond err can all reach the precision 0 09 but it seems that they do not depend on the matrix size n very much. owever the CPU time grows quickly as n increases. In Table welistthecputimeε X ε PQ and Cond err respectively. 6. Conclusion In this paper we consider () with two special constraints PX = sxq and SX = XR wherep Q C n n are ermitian involutory S R C n n are ermitian idempotent and s = ±. We represent the general solutions to the constrained equation by eigenvalue decompositions of P Q S R release the involved eigenvector by Moore-Penrose generalizedinversesandgettheeigenvector-freeformulasof the general solutions. Acknowledgments The author is grateful to the referees for their enlightening suggestions. Moreover the research was supported in part by the Natural Science Foundation of Zhejiang Province and National Natural Science Foundation of China (Grant nos. Y60639 LQA007 and 04). References [].-C. Chen Generalized reflexive matrices: special properties and applications SIAM Journal on Matrix Analysis and Applicationsvol.9no.pp [].-C. Chen and A.. Sameh A matrix decomposition method for orthotropic elasticity problems SIAM Journal on Matrix Analysis and Applicationsvol.0no.pp [3] F. Li X. u and L. Zhang The generalized reflexive solution foraclassofmatrixequations(ax = B XC = D) Acta Mathematica Scientia Bvol.8no.pp [4] C.MengX.uandL.Zhang Theskew-symmetricorthogonal solutions of the matrix equation AX = B Linear Algebra and its Applicationsvol.40pp [5]C.J.MengandX.Y.u Aninverseproblemforsymmetric orthogonal matrices and its optimal approximation Mathematica Numerica Sinica vol. 8 no. 3 pp [6] Z.-Y. Peng The inverse eigenvalue problem for ermitian antireflexive matrices and its approximation Applied Mathematics and Computationvol.6no.3pp [7] Z.-Y. Peng and X.-Y. u The reflexive and anti-reflexive solutions of the matrix equation AX = B Linear Algebra and its Applicationsvol.375pp [8]Y.QiuZ.ZhangandJ.Lu ThematrixequationsAX = B XC = D with PX = sxp constraint Applied Mathematics and Computationvol.89no.pp [9] Q.-W. Wang S.-W. Yu and C.-Y. Lin Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications Applied Mathematics and Computationvol.95no.pp [0] Q.-W. Wang.-X. Chang and C.-Y. Lin P-(skew)symmetric common solutions to a pair of quaternion matrix equations Applied Mathematics and Computationvol.95no.pp [] Q.-W. Wang J. W. van der Woude and.-x. Chang A system of real quaternion matrix equations with applications Linear Algebra and its Applicationsvol.43no.pp [] Q.-W. Wang and Z.-. e Some matrix equations with applications Linear and Multilinear Algebra vol.60no.-pp [3]Q.WangandZ.e Asystemofmatrixequationsandits applications Science China. Mathematics vol. 56 no. 9 pp [4] Z.-. e and Q.-W. Wang A real quaternion matrix equation with applications Linear and Multilinear Algebra vol. 6no. 6 pp [5] K. E. Chu Singular value and generalized singular value decompositions and the solution of linear matrix equations Linear Algebra and its Applications vol. 88/89 pp [6] K. E. Chu Symmetric solutions of linear matrix equations by matrix decompositions Linear Algebra and its Applicationsvol. 9 pp

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