Open Access
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 1 X 1 B 1 + A 2 X 2 B 2 = E , C 1 X 1 D 1 + C 2 X 2 D 2 = F with real matrices X 1 and X 2 . By this iterative algorithm, the solvability of the system can be determined automatically. When the system is consistent, for any initial matrices X 1 0 and X 2 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 X ^ 1 and X ^ 2 to the given matrices X ~ 1 and X ~ 2 in Frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations A 1 X ¯ 1 B 1 + A 2 X ¯ 2 B 2 = E ¯ , C 1 X ¯ 1 D 1 + C 2 X ¯ 2 D 2 = F ¯ , where E ¯ = E - A 1 X ~ 1 B 1 - A 2 X ~ 2 B 2 , F ¯ = F - C 1 X ~ 1 D 1 - C 2 X ~ 2 D 2 . The given numerical example demonstrates that the iterative algorithm is efficient. Especially, when the numbers of the parameter matrices A 1 , A 2 , B 1 , B 2 , C 1 , C 2 , D 1 , D 2 are large, our algorithm is efficient as well.

Citation

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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

Rights: Copyright © 2013 Hindawi

Vol.2013 • No. SI33 • 2013
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