an iterative method for the skew symmetric solution and the optimal approximate solution of the matrix equation axb c in this paper an iterative method is constructed to solve the linear matrix equation axb c over skew symmetric matrix x by the iterative method the solvability of the equation axb c over skew symmetric matrix can be determined automatically when the equation axb c is consistent over skew symmetric matrix x for any skew symmetric initial iterative matrix x 1 the skew symmetric solution can be obtained within finite iterative steps in the absence of roundoff errors the unique least norm skew symmetric iterative solution of axb c can be derived when an appropriate initial iterative matrix is chosen a sufficient and necessary condition for whether the equation axb c is inconsistent is given furthermore the optimal approximate solution of axb c for a given matrix x 0 can be derived by finding the least norm skew symmetric solution of a new corresponding matrix equation email protected email protected finally several numerical examples are given to illustrate that our iterative method is effective

the reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a reflexive or anti reflexive matrix with respect to the generalized reflection matrix p if a pap or a pap this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti reflexive with respect to a generalized reflection matrix p solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

singular value and generalized singular value decompositions and the solution of linear matrix equations abstract the solution of the linear matrix equations i axb cyd e and ii axb fxg e h are considered new necessary and sufficient conditions for the consistency of the equations are derived some using the generalized singular value decomposition special cases iii ax yd e and iv axb e are treated using the singular value decomposition numerical algorithms for the solutions are also suggested

an iterative method for the least squares symmetric solution of the linear matrix equation axb c abstract this paper an iterative method is presented to solve the minimum frobenius norm residual problem min axb c with unknown symmetric matrix x by this iterative method for any initial symmetric matrix x0 a solution x can be obtained within finite iteration steps in the absence of roundoff errors and the solution x with least norm can be obtained by choosing a special kind of initial symmetric matrix in addition the unique optimal approximation solution x to a given matrix x in frobenius norm can be obtained by first finding the least norm solution x of the new minimum residual problem min a x b c with unknown symmetric matrix x where c c a x x t 2 b given numerical examples are show that the iterative method is quite efficient

positive definite solutions of the nonlinear matrix equation x a h x 1 a i this paper is concerned with the positive definite solutions to the nonlinear matrix equation x a email protected 1a i where x is the unknown and a is a given complex matrix by introducing and studying a matrix operator on complex matrices it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation w b tw 1b i which has been extensively studied in the literature where b is a real matrix and is uniquely determined by a it is also shown that if the considered nonlinear matrix equation has a positive definite solution then it has the maximal and minimal solutions bounds of the positive definite solutions are also established in terms of matrix a finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed

computing thecs decomposition of a partitioned orthonormal matrix this paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns

an iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation axb c an iteration method is constructed to solve the linear matrix equation axb c over symmetric x by this iteration method the solvability of the equation axb c over symmetric x can be determined automatically when the equation axb c is consistent over symmetric x its solution can be obtained within finite iteration steps and its least norm symmetric solution can be obtained by choosing a special kind of initial iteration matrix furthermore its optimal approximation solution to a given matrix can be derived by finding the least norm symmetric solution of a new matrix equation axb c finally numerical examples are given for finding the symmetric solution and the optimal approximation symmetric solution of the matrix equation axb c

the symmetric solution of the matrix equations ax ya c axat bybt c and atxa btxb c d abstract the necessary and sufficient conditions for the existence of and the expressions for the symmetric solutions of matrix equations i ax ya c ii axa t byb t c and iii a t xa b t xb c d are derived in addition the minimum 2 norm least squares symmetric solution of equation i the minimum 2 norm symmetric solution of equation ii and the least squares solution of equation iii are obtained

the p q generalized reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a p q generalized reflexive or anti reflexive matrix with respect to the generalized reflection matrix dual p q if a paq or a paq this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the p q generalized reflexive and anti reflexive solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm has been provided

an iterative algorithm for the reflexive solutions of the generalized coupled sylvester matrix equations and its optimal approximation abstract the generalized coupled sylvester matrix equations ay zb cy zd e f with unknown matrices y z are encountered in many systems and control applications also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils in this work we construct an iterative algorithm to solve the generalized coupled sylvester matrix equations over reflexive matrices y z and when the matrix equations are consistent for any initial matrix pair y 0 z 0 a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors and the least frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair also we obtain the optimal approximation reflexive solution pair to a given matrix pair y z in the reflexive solution pair set of the generalized coupled sylvester matrix equations ay zb cy zd e f moreover several numerical examples are given to show the efficiency of the presented iterative algorithm

new matrix iterative methods for constraint solutions of the matrix equation axb c in this paper two new matrix iterative methods are presented to solve the matrix equation axb c the minimum residual problem min x email protected email protected and the matrix nearness problem min x s email protected x x where s is the set of constraint matrices such as symmetric symmetric r symmetric and r s symmetric and s e is the solution set of above matrix equation or minimum residual problem these matrix iterative methods have faster convergence rate and higher accuracy than the matrix iterative methods proposed in deng et al 2006 13 huang et al 2008 15 peng 2005 16 and lei and liao 2007 17 paige s algorithms are used as the frame method for deriving these matrix iterative methods numerical examples are used to illustrate the efficiency of these new methods

symmetric solutions of linear matrix equations by matrix decompositions abstract the symmetric solutions of linear matrix equations like ax b have been considered using the kronecker product formulation 17 and the commutation matrix k 5 11 in this paper a novel approach is followed using the singular value generalized singular value real schur and real generalized schur decompositions solvability conditions for the symmetric and nonsymmetric solutions are obtained and the numerical properties of the related solution procedures are discussed briefly

the solution of the matrix equations axb cxd e and ya dz yc bz e f the conditions for the existence of a unique solution of the matrix equation axb cxd e are proved to be that i the pencils a c and d b are regular and ii the spectra of the pencils have an empty intersection a numerical algorithm for solving the equation is proposed the possibility of a least squares type solution is briefly discussed the set of equations ya dz yc bz e f is proved to be equivalent to the aforementioned equation and its solution is also investigated a numerical algorithm is proposed

the hermitian reflexive solutions to the matrix inverse problem ax b in this paper the necessary and sufficient conditions for the solvability of the matrix inverse problem ax b in hermitian reflexive matrix set hj n x n are given and the solution expression of this matrix inverse problem is also presented moreover an optimal approximation to any matrix in the above solution set in frobenius norm is provided

linear matrix equations from an inverse problem of vibration theory abstract the symmetric positive semidefinite and positive definite real solutions of matrix equations a t xa d and a t xa xa y ad d 0 are considered necessary and sufficient conditions for the existence of such solutions and their general forms are derived using the singular value decomposition the theory is motivated and illustrated with a problem of vibration theory

on the symmetric solutions of a linear matrix equation a formula for the partitioned minimum norm reflexive generalized inverse is applied to find the general symmetric solution x to the matrix equation ax b also the dimension of the space of symmetric solutions is established

some boundary layer growth problems associated with a flat plate with suction or injection in this sequel to a recent work of s turbatu bull math soc sci math r s roumanie n s 40 88 1997 183 192 the authors further investigate and obtain exact analytical solutions of the navier stokes equations for the boundary layer growth on an infinite porous flat plate with uniform suction or blowing assuming the normal velocity v t to be a constant several different cases of the velocity u t along the plate are treated here in a unified manner

common hermitian and positive solutions to the adjointable operator equations ax c xb d let a be a c algebra for any hilbert a modules h and k let l h k be the set of adjointable operators from h to k let h k l be hilbert a modules a c l h k and b d l l h in this paper we propose necessary and sufficient conditions for the existence of common hermitian and positive solutions x l h to the equations ax c xb d and obtain the formulae for the general forms of these solutions some results known for finite matrices and hilbert space operators are extended to the adjointable operators acting on hilbert c modules

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar