An efficient algorithm for the least-squares reflexive solution of the matrix equation A1XB1 = C1, A2XB2 = C2

Abstract

A n × n real matrix P is said to be a real generalized reflection matrix if P T = P and P 2 = I. A n × n real matrix X is said to be a reflexive matrix with respect to the generalized reflection matrix P if X = PXP. In this paper, the concept of gradient matrix (∇ F( X)) is presented and an algorithm is constructed to solve the reflexive with respect to the generalized reflection matrix P solution of the minimum Frobenius norm residual problem: A 1 XB 1 A 2 XB 2 - C 1 C 2 = min . By this algorithm, for any initial reflexive matrix X 1, 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 reflexive matrix. In addition, in the solution set of above problem, the unique optimal approximation solution X ^ to a given matrix X 0 in Frobenius norm can be obtained by finding the least-norm reflexive solution X ∼ ∗ of the new minimum residual problem: min A 1 X ∼ B 1 A 2 X ∼ B 2 - C 1 ∼ C 2 ∼ , where C 1 ∼ = C 1 - A 1 X 0 B 1 , C 2 ∼ = C 2 - A 2 X 0 B 2 . Given numerical examples show that the iterative method is quite efficient.