The symmetric minimal rank solution of the matrix equation AX=B and the optimal approximation

Abstract

By applying the matrix rank method, the set of symmetric matrix solutions with prescribed rank to the matrix equation AX = B is found. An expression is provided for the optimal approximation to the set of the minimal rank solutions. 1. Introduction. We first introduce some notation to be used. Let C n×m de- note the set of all n × m complex matrices; R n×m denote the set of all n × m real matrices; SR n×n and ASR n×n be the sets of all n × n real symmetric and antisym- metric matrices respectively; OR n×n be the sets of all n × n orthogonal matrices. The symbols A T , A + , A − , R(A), N (A)and r(A)stand, respectively, for the transpose, Moore-Penrose generalized inverse, any generalized inverse, range (column space), null space and rank of A ∈ R n×m .T he symbolsEA and FA stand for the two projec- tors EA = I − AA − and FA = I − A − A induced by A. The matrices I and 0 denote, respectively, the identity and zero matrices of sizes implied by the context. We use =t race(B T A)to define the inner product of matrices A and B in R n×m . Then R n×m is an inner product Hilbert space. The norm of a matrix generated by the inner product is the Frobenius norm �·� ,t hat is,� A� = √ = (trace(A T A)) 1 2 .