2013 Iterative Solution to a System of Matrix Equations
Yong Lin, Qing-Wen Wang
Abstr. Appl. Anal. 2013(SI33): 1-7 (2013). DOI: 10.1155/2013/124979

## 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 ${\stackrel{^}{X}}_{\mathrm{1}}$ and ${\stackrel{^}{X}}_{\mathrm{2}}$ to the given matrices ${\stackrel{~}{X}}_{\mathrm{1}}$ and ${\stackrel{~}{X}}_{\mathrm{2}}$ 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 $\overline{E}=E-{A}_{\mathrm{1}}{\stackrel{~}{X}}_{\mathrm{1}}{B}_{\mathrm{1}}-{A}_{\mathrm{2}}{\stackrel{~}{X}}_{\mathrm{2}}{B}_{\mathrm{2}}$, $\overline{F}=F-{C}_{\mathrm{1}}{\stackrel{~}{X}}_{\mathrm{1}}{D}_{\mathrm{1}}-{C}_{\mathrm{2}}{\stackrel{~}{X}}_{\mathrm{2}}{D}_{\mathrm{2}}$. 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.

## Citation

Yong Lin. Qing-Wen Wang. "Iterative Solution to a System of Matrix Equations." Abstr. Appl. Anal. 2013 (SI33) 1 - 7, 2013. https://doi.org/10.1155/2013/124979

## Information

Published: 2013
First available in Project Euclid: 26 February 2014

zbMATH: 1297.65046
MathSciNet: MR3121495
Digital Object Identifier: 10.1155/2013/124979 