The Re-nonnegative definite solutions to the matrix inverse problem \(AX=B\)
type: Item
instance of: scholarly article
title: The Re-nonnegative definite solutions to the matrix inverse problem \(AX=B\)
zbMATH Open document ID: 0851.15004
DOI: 10.1016/0024-3795(94)00142-1
author: Wu Lei
author: Bryan E. Cain
published in: Linear Algebra and its Applications
publication date: 1996-11-26T00:00:00Z
review text: Let \(A\) be an \(n\times n\) complex matrix. Then \(A\) is called ``Re-nonnegative definite'' (Re-nnd) if the real part of \(x* Ax\) is nonnegative for all complex \(n\)-vectors \(x\) (equivalently, if \(H(A):= {1\over 2} (A+ A*)\) is nonnegative definite). Analogous conditions define ``Re-positive definite matrices'' (Re-pd). The primary result of the paper is a necessary and sufficient condition for there to be at least one Re-nnd matrix \(A\) satisfying \(AX =B\) for given \(n\times m\) matrices \(X\) and \(B\). Theorem 5. An \(n\times n\) Re-nnd matrix \(A\) exists such that \(AX= B\) if and only if (i) there exists an \(n\times n\) matrix \(A_0\) such that \(A_0 X=B\) and (ii) \(X * B\) is Re-nnd. This result is analogous to a theorem for Re-pd matrices proved earlier by \textit{L. Wu} [Linear Algebra Appl. 174, 145-151 (1992; Zbl 0754.15003)].
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 871263
zbMATH Keywords: Re-nonnegative definite solutions
zbMATH Keywords: matrix inverse problem
zbMATH Keywords: matrix equation
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00142-1
OpenAlex ID: W4210712975
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cites work: The Re-positive definite solutions to the matrix inverse problem \(AX = B\)
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description: scientific article; zbMATH DE number 871263