The \((P,Q)\)-(skew)symmetric extremal rank solutions to a system of quaternion matrix equations
type: Item
instance of: scholarly article
title: The \((P,Q)\)-(skew)symmetric extremal rank solutions to a system of quaternion matrix equations
zbMATH Open document ID: 1222.15022
author: Qin Zhang
author: Qing-Wen Wang
published in: Applied Mathematics and Computation
publication date: 2011-06-27T00:00:00Z
review text: The paper deals with extremal rank solutions to a system of quaternion matrix equations. \(H^{m \times n}\) denotes the set of \(m \times n\) matrices over the real quaternion algebra \[ H=\{a_0+a_1i+a_2j+a_3k \;: \;i^2=j^2=k^2=ijk=-1 \;\text{and} \;a_0,a_1,a_2,a_3 \;\text{are real numbers} \}. \] A matrix \(A \in H^{m \times n}\) is called \((P,Q)\)-symmetric (or \((P,Q)\)-skewsymmetric) if \(A=PAQ\) (or \(A=-PAQ\)), where \(P \in H^{m \times m}\) and \(Q \in H^{n \times n}\) are involution matrices. Consider the system of matrix equations over \(H\) \[ AX=B, \;XC=D. \tag{*} \] In this work, the authors analyze the \((P,Q)\)-(skew)symmetric maximal and minimal rank solutions of this system. They obtain necessary and sufficient conditions for the existence of \((P,Q)\)-symmetric and \((P,Q)\)-skewsymmetric solutions to the above system and give the expressions of such solutions when the solvability conditions are satisfied. The authors also establish formulas of maximal and minimal ranks of \((P,Q)\)-symmetric and \((P,Q)\)-skewsymmetric solutions of (*) and derive the expressions of \((P,Q)\)-(skew)symmetric maximal and minimal rank solutions of (*). Finally, the authors present a numerical example that confirms the theoretical results obtained.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 5913740
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: \((P,Q)\)-symmetric matrix
zbMATH Keywords: \((P,Q)\)-skewsymmetric matrix
zbMATH Keywords: minimal rank
zbMATH Keywords: maximal rank
zbMATH Keywords: extremal rank solutions
zbMATH Keywords: system of quaternion matrix equations
zbMATH Keywords: quaternion algebra
zbMATH Keywords: numerical example
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: j.amc.2011.04.011
OpenAlex ID: W1977502914
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DOI: 10.1016/J.AMC.2011.04.011
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description: scientific article; zbMATH DE number 5913740