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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Bibliographic Details
Published in:Linear algebra and its applications Vol. 431; no. 12; pp. 2291 - 2303
Main Authors: Wang, Qing-Wen, van der Woude, J.W., Chang, Hai-Xia
Format: Journal Article
Language:English
Published: Amsterdam Elsevier Inc 01.12.2009
Elsevier
Subjects:
[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 Access:Get full text
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Summary: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