An iterative method for the least squares symmetric solution of matrix equation $AXB = C$

Abstract

This paper an iterative method is presented to solve the minimum Frobenius norm residual problem: \(\min\|AXB-C\|\) with unknown symmetric matrix \(X\). By the iterative method, for any initial symmetric matrix \(X_{0}\), a solution \(X^{*}\) can be obtained within finite iteration steps in the absence of roundoff errors, and the solution \(X^{*}\) with least Frobenius norm can be obtained by choosing a special kind of initial symmetric matrix. In addition, in the solution set of the minimum Frobenius norm residual problem, the unique optimal approximation solution \({\hat {X}}\) to a given matrix \(\overline{X}\) in Frobenius norm can be expressed as \({\hat {X}}=X^{*}+\frac{\overline{X}+\overline{X}^{T}}{2}\), where \(X^{*}\) is the least norm symmetric solution of the new minimum residual problem: \(\min\|AXB-C\|\) with \(C=C-A\overline{X}B\). Given numerical examples are show that the iterative method is quite efficient.