Abstract and Applied Analysis

Iterative Solution to a System of Matrix Equations

Yong Lin and Qing-Wen Wang

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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.

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


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