Further results on invariance of the eigenvalues of matrix product involving generalized inverses
type: Item
instance of: scholarly article
title: Further results on invariance of the eigenvalues of matrix product involving generalized inverses
zbMATH Open document ID: 0851.15006
DOI: 10.1016/0024-3795(94)00001-8
author: Jerzy K. Baksalary
author: Augustyn Markiewicz
published in: Linear Algebra and its Applications
publication date: 1996-11-26T00:00:00Z
review text: Let \(L\) be an \(m\times n\) matrix over \(\mathbb{C}\), and let \(L*\) denote its Hermitian conjugate. The generalized inverses of \(L\) are the \(n\times m\) matrices \(L^-\) such that \(LL^- L=L\). Let \(A\), \(B\) and \(C\) denote respectively \(m\times n\), \(p\times n\) and \(p\times m\) matrices over \(\mathbb{C}\). It is known [\textit{J. K. Baksalary} and \textit{T. Pukkila}, Linear Algebra Appl. 165, 125-130 (1992; Zbl 0743.15005)] that the set of eigenvalues of \(AB^- C\) is independent of the choice of \(B^-\) if and only if: (i) \(\text{range} (A* C*) \subseteq \text{range} (B*)\) and \(\text{range} (CA) \subseteq \text{range} (B)\). The main result (Theorem 1) of the present paper is that: if (i) does not hold then for each \(\lambda\in \mathbb{C}\) there is a generalized inverse \(B^-\) such that \(\lambda\) is an eigenvalue of \(AB^- C\).
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 885063
zbMATH Keywords: matrix product
zbMATH Keywords: generalized inverses
zbMATH Keywords: eigenvalues
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00001-8
OpenAlex ID: W2092380874
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description: scientific article; zbMATH DE number 885063