A system of real quaternion matrix equations with applications
type: Item
instance of: scholarly article
title: A system of real quaternion matrix equations with applications
zbMATH Open document ID: 1180.15019
author: Haixia Chang
author: Qing-Wen Wang
author: Jacob W. van der Woude
published in: Linear Algebra and its Applications
publication date: 2009-11-23T00:00:00Z
review text: A system of general linear matrix equations over the real quaternions is studied and solvability conditions with explicit expressions of the solutions, when the matrix system is solvable, are given. Let \({\mathbb H}^{n \times m}\) stands for the set of all \(n \times m\) matrices over the real quaternion algebra. A matrix \(A \in {\mathbb H}^{n \times m}\) is called \((P,Q)\)-symmetric or \((P,Q)\)-skew-symmetric if \(A=PAQ\) or \(A= -PAQ\), respectively, where \(P \in {\mathbb H}^{n \times n}\), \(Q \in {\mathbb H}^{m \times m}\) are involutory matrices, i. e., \(P^2 = I\), \(Q^2 = I\). Firstly, some new necessary and sufficient conditions for the existence of a solution to the system \[ \begin{aligned} A_1 X_1 &= C_1, \qquad A_2 X_2 = C_3,\\ X_1 B_1 &= C_2, \qquad X_2 B_2 = C_4, \qquad A_3 X_1 B_3 + A_4 X_2 B_4 = C_5 \end{aligned} \] over \({\mathbb H}\) are established and an explicit expression of the general solution is derived. Some auxiliary results on certain related systems over \({\mathbb H}\) are also given. Then, as application, necessary and sufficient conditions for the existence of the \((P, Q)\)-symmetric solution to the system \[ A_a X = C_a, \qquad X B_b = C_b, \qquad A_c X B_c = C_c \] over \({\mathbb H}\), which was first investigated by \textit{P. Bhimasankaram} [Sankhyā Ser. A 38, 404--409 (1976; Zbl 0411.15008)], are given. Finally, an algorithm and a numerical example to illustrate the presented results are shown.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 5637025
zbMATH Keywords: system of matrix equations
zbMATH Keywords: quaternion matrix
zbMATH Keywords: inner inverse of a matrix
zbMATH Keywords: reflexive inverse of a matrix
zbMATH Keywords: Moore-Penrose inverse of a matrix
zbMATH Keywords: \((P,Q)\)-symmetric matrix
zbMATH Keywords: algorithm
zbMATH Keywords: numerical example
MaRDI profile type: Publication
full work available at URL: j.laa.2009.02.010
OpenAlex ID: W1980549954
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DOI: 10.1016/J.LAA.2009.02.010
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description: scientific article; zbMATH DE number 5637025