Ranks of Hermitian and skew-Hermitian solutions to the matrix equation \(AXA^*=B\)
type: Item
instance of: scholarly article
title: Ranks of Hermitian and skew-Hermitian solutions to the matrix equation \(AXA^*=B\)
zbMATH Open document ID: 1180.15018
author: Yongge Tian
author: Yoshio Takane
author: Yonghui Liu
published in: Linear Algebra and its Applications
publication date: 2009-11-23T00:00:00Z
review text: Extremal ranks of solutions of some complex matrix equations and their applications are analyzed and discussed. The authors consider the following problems on solutions to the equation \(AXA^* = B\), where \(A \in {\mathbb C}^{m \times n}\) and \(B \in {\mathbb C}^{m \times m}\) are given with \(B^* = \pm B\) and \(X \in {\mathbb C}^{n \times n}\) is the unknown matrix: {\parindent=6mm \begin{itemize}\item[(i)] The maximal and minimal ranks of Hermitian and skew-Hermitian solutions \(X\), when \(B^* = B\) and \(B^* = -B\), respectively. \item[(ii)] The maximal and minimal ranks of the two real matrices \(X_0\) and \(X_1\) in Hermitian and skew-Hermitian solutions \(X = X_0 + iX_1\), when \(B^* = B\) and \(B^* = -B\), respectively. \end{itemize}} As applications, the maximal and minimal ranks of the real matrices \(C\) and \(D\) in a Hermitian (skew-Hermitian) \(g\)-inverse \((A + iB)^- = (C + iD)\) of a Hermitian (skew-Hermitian) matrix \(A + iB\) are shown.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 5637031
zbMATH Keywords: solvability conditions
zbMATH Keywords: Hermitian solution
zbMATH Keywords: skew-Hermitian solution
zbMATH Keywords: maximal rank
zbMATH Keywords: minimal rank
zbMATH Keywords: generalized inverse
zbMATH Keywords: complex matrix equations
MaRDI profile type: Publication
full work available at URL: j.laa.2009.03.011
OpenAlex ID: W2028242709
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DOI: 10.1016/J.LAA.2009.03.011
description: scientific article; zbMATH DE number 5637031