Iterative Solution to a System of Matrix Equations

Abstract

An efficient iterative algorithm is presented to solve a system of linear matrix equations $A_{\mathrm{1}}{X}_{\mathrm{1}}{B}_{\mathrm{1}}+A_{\mathrm{2}}{X}_{\mathrm{2}}{B}_{\mathrm{2}}=E$, $C_{\mathrm{1}}{X}_{\mathrm{1}}{D}_{\mathrm{1}}+C_{\mathrm{2}}{X}_{\mathrm{2}}{D}_{\mathrm{2}}=F$ with real matrices $X_{\mathrm{1}}$ and $X_{\mathrm{2}}$. By this iterative algorithm, the solvability of the system can be determined automatically. When the system is consistent, for any initial matrices $X_{\mathrm{1}}^{\mathrm{0}}$ and $X_{\mathrm{2}}^{\mathrm{0}}$, a solution can be obtained in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix. In addition, the unique optimal approximation solutions ${\widehat{X}}_{\mathrm{1}}$ and ${\widehat{X}}_{\mathrm{2}}$ to the given matrices ${\tilde{X}}_{\mathrm{1}}$ and ${\tilde{X}}_{\mathrm{2}}$ in Frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations $A_{\mathrm{1}}{\overline{X}}_{\mathrm{1}}{B}_{\mathrm{1}}+A_{\mathrm{2}}{\overline{X}}_{\mathrm{2}}{B}_{\mathrm{2}}=\overline{E},C_{\mathrm{1}}{\overline{X}}_{\mathrm{1}}{D}_{\mathrm{1}}+C_{\mathrm{2}}{\overline{X}}_{\mathrm{2}}{D}_{\mathrm{2}}=\overline{F}$, where $\overline{E}=E-A_{\mathrm{1}}{\tilde{X}}_{\mathrm{1}}{B}_{\mathrm{1}}-A_{\mathrm{2}}{\tilde{X}}_{\mathrm{2}}{B}_{\mathrm{2}}$, $\overline{F}=F-C_{\mathrm{1}}{\tilde{X}}_{\mathrm{1}}{D}_{\mathrm{1}}-C_{\mathrm{2}}{\tilde{X}}_{\mathrm{2}}{D}_{\mathrm{2}}$. The given numerical example demonstrates that the iterative algorithm is efficient. Especially, when the numbers of the parameter matrices $A_{\mathrm{1}},A_{\mathrm{2}},B_{\mathrm{1}},B_{\mathrm{2}},C_{\mathrm{1}},C_{\mathrm{2}},D_{\mathrm{1}},D_{\mathrm{2}}$ are large, our algorithm is efficient as well.