Abstract and Applied Analysis

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 ${\stackrel{^}{X}}_{\mathrm{1}}$ 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.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 124979, 7 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393447685

Digital Object Identifier
doi:10.1155/2013/124979

Mathematical Reviews number (MathSciNet)
MR3121495

Zentralblatt MATH identifier
1297.65046

Citation

Lin, Yong; Wang, Qing-Wen. Iterative Solution to a System of Matrix Equations. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 124979, 7 pages. doi:10.1155/2013/124979. https://projecteuclid.org/euclid.aaa/1393447685