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

摘要

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 $C=C-Aoverline{X}B$ . Given numerical examples are show that the iterative method is quite efficient.