A system of real quaternion matrix equations with applications

Let H be the real quaternion algebra and H n × m denote the set of all n × m matrices over H . Let P ∈ H n × n and Q ∈ H m × m be involutions, i.e., P 2 = I , Q 2 = I . A matrix A ∈ H n × m is said to be ( P , Q ) -symmetric if A = PAQ . This paper studies the system of linear real quaternion matrix...

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Veröffentlicht in:	Linear algebra and its applications Jg. 431; H. 12; S. 2291 - 2303
Hauptverfasser:	Wang, Qing-Wen, van der Woude, J.W., Chang, Hai-Xia
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Amsterdam Elsevier Inc 01.12.2009 Elsevier
Schlagworte:	[formula omitted]-symmetric matrix Algebra Exact sciences and technology Inner inverse of a matrix Linear and multilinear algebra, matrix theory Mathematics Moore–Penrose inverse of a matrix Quaternion matrix Reflexive inverse of a matrix Sciences and techniques of general use System of matrix equations Quaternion matrix Moore–Penrose inverse of a matrix 15A24 15A57 15A33 15A06 Reflexive inverse of a matrix Inner inverse of a matrix 15A09 System of matrix equations ( P , Q ) -symmetric matrix Existence of solution Moore Penrose inverse Quaternion Moore-Penrose inverse of a matrix Generalized inverse Commutativity Algebra Inverse matrix Matrix equation Real matrix Operator equation (P, Q)-symmetric matrix Involution Linear system Matrix inversion Symmetric matrix Ring Solvability
ISSN:	0024-3795
Online-Zugang:	Volltext
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Zusammenfassung:	Let H be the real quaternion algebra and H n × m denote the set of all n × m matrices over H . Let P ∈ H n × n and Q ∈ H m × m be involutions, i.e., P 2 = I , Q 2 = I . A matrix A ∈ H n × m is said to be ( P , Q ) -symmetric if A = PAQ . This paper studies the system of linear real quaternion matrix equations A 1 X 1 = C 1 X 1 B 1 = C 2 A 2 X 2 = C 3 X 2 B 2 = C 4 A 3 X 1 B 3 + A 4 X 2 B 4 = C c . We present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied. As applications, we discuss the necessary and sufficient conditions for the system A a X = C a , XB b = C b , A c XB c = C c to have a ( P , Q ) -symmetric solution. We also show an expression of the ( P , Q ) -symmetric solution to the system when the solvability conditions are met. Moreover, we provide an algorithm and a numerical example to illustrate our results. The findings of this paper extend some known results in the literature.
ISSN:	0024-3795
DOI:	10.1016/j.laa.2009.02.010