Least - Squares Mirrorsymmetric Solution for Matrix Equations ( AX = B , XC = D )

Matrix equations (AX = B,XC = D) is a class of important matrix equations. The problem of the solution of the matrix equations (AX = B,XC = D) arise in engineering and in some special matrix inverse problems [1-3]. Many authors have been devoted to the study of this problem, and a series of useful results have been obtained. For example, Mitra [4] gave the common solution of minimum possible rank by using generalized inverse of matrix. Chu [5], Mitra [6] presented the necessary and sufficient conditions for the solvability and general solution by using the singular value decomposition (SVD) and generalized inverse of matrix, respectively. When the solution matrix X is constrained and the matrix equations are not consistent, it is necessary to study the least-squares constrained solution of the corresponding matrix equations. The purpose of this paper is to discuss the least-squares mirrorsymmetric solution of the matrix equations (AX = B,XC = D) by using the special structure of mirrorsymmetric matrices. The background for introducing the definition of mirrorsymmetric matrices is to study odd/even-mode decomposition of symmetric multiconductor transmission lines (MTL)[7]. We now introduce some notation. Let R be the set of all n × m real matrices; R r be the set of all matrices in R with rank r, R(A) denote the rank of A; OR denote