The Solutions Of A Few Constrained Matrix Equations And Their Optimal Approximations

Constrained matrix equation problem is to find the solution of a matrix equation in a constrained matrix set. In this paper, we introduce the concepts and structures of (M,N)-symmetric matrix and the generalized reflexive matrix. We use the general-ized singular-value decomposition, Kroneker product and Moore-Penrose generalized inverse of matrices to analysis the problem and to derive the necessary and sufficient conditions for the existence of and the expressions for the (M, N)-symmetric solutions of the matrix equation AXB= C and matrix equations AX= B, CXD= E. The related optimal approximation problem to a given matrix on the solution set is solved. And we derive the necessary and sufficient conditions and expression for generalized reflexive solutions of matrix equations AX= B,CXD= E by using singular-value decomposition and generalized singular-value decomposition. Moreover, the optimal approximation is provided.