Nearest southeast submatrix that makes multiple a prescribed eigenvalue.
type: Item
instance of: scholarly article
title: Nearest southeast submatrix that makes multiple a prescribed eigenvalue.
zbMATH Open document ID: 1194.15024
author: J. Martínez
published in: Linear Algebra and its Applications
publication date: 2009-02-10T00:00:00Z
review text: Let \(A\in \mathbb C^{n\times n}\), \(B\in \mathbb C^{n\times m}\), \(C\in \mathbb C^{m\times n}\), \(D\in \mathbb C^{m\times m}\) be given complex matrices and \(z_0\in \mathbb C\). Let \(\mathcal X\) be the set of all \(X\in \mathbb C^{m\times m}\) such that \(z_0\) is a multiple eigenvalue of \(\left(\begin{smallmatrix} A&B\\ C&X\end{smallmatrix}\right)\). The question: what is \(\text{dist}(D,\mathcal X)\) is investigated. The problem is first reduced to the case \(z_0=0\). It is proved that if \(z_0=0\) is not an eigenvalue of \(A\), then \(\text{dist}(D,\mathcal X)=\sup_{t\in\mathbb R}\sigma_{2m-1}(S_2(t,D))\) where \(S_2(t,D)\) is a \(2m\times2m\) matrix (its construction is too complicated to be stated here) and \(\sigma_{2m-1}(S_2(t,D))\) denotes its \((2m-1)\)th (in decreasing order) singular value. In the case when \(z_0\) is an eigenvalue of \(A\) an interesting conjecture is stated.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 93B10
zbMATH DE Number: 5505623
zbMATH Keywords: nearest matrix
zbMATH Keywords: multiple eigenvalue
zbMATH Keywords: Wilkinson's problem
zbMATH Keywords: controllable
zbMATH Keywords: observable
zbMATH Keywords: derivatives of singular values
reviewed by: Michal Zajac
MaRDI profile type: Publication
full work available at URL: j.laa.2008.10.012
OpenAlex ID: W2083230866
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DOI: 10.1016/J.LAA.2008.10.012
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description: scientific article; zbMATH DE number 5505623