The general coupled matrix equations over generalized bisymmetric matrices

Abstract

In the present paper, by extending the idea of conjugate gradient (CG) method, we construct an iterative method to solve the general coupled matrix equations

$$\sum _{j=1}^p{A}_{\mathit{ij}}{X}_j{B}_{\mathit{ij}}=M_i\text{,}i=1\text{,}2\text{,}\dots \text{,}p\text{,}$$

(including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over generalized bisymmetric matrix group $(X_1\text{,}{X}_2\text{,}\dots \text{,}{X}_p)$. By using the iterative method, the solvability of the general coupled matrix equations over generalized bisymmetric matrix group can be determined in the absence of roundoff errors. When the general coupled matrix equations are consistent over generalized bisymmetric matrices, a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm generalized bisymmetric solution group of the general coupled matrix equations can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation generalized bisymmetric solution group to a given matrix group $({\widehat{X}}_1\text{,}{\widehat{X}}_2\text{,}\dots \text{,}{\widehat{X}}_p)$ in Frobenius norm can be obtained by finding the least Frobenius norm generalized bisymmetric solution group of new general coupled matrix equations. The numerical results indicate that the iterative method works quite well in practice.