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

A n × n real matrix P is said to be a real generalized reflection matrix if PT = P and P2 = 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:A1XB1A2XB2-C1C2=min. By this algorithm, for any initial reflexive matrix X1, 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 X0 in Frobenius norm can be obtained by finding the least-norm reflexive solution X∼∗ of the new minimum residual problem: minA1X∼B1A2X∼B2-C1∼C2∼, where C1∼=C1-A1X0B1, C2∼=C2-A2X0B2. Given numerical examples show that the iterative method is quite efficient.