Order invariant spectral properties for several matrices
type: Item
instance of: scholarly article
title: Order invariant spectral properties for several matrices
zbMATH Open document ID: 1187.15011
author: S. Furtado
author: Charles R. Johnson
published in: Linear Algebra and its Applications
publication date: 2010-03-04T00:00:00Z
full work available at URL: https://scholarworks.wm.edu/aspubs/1238
review text: Given a set \(\left\{ A_{1},\dots,A_{m}\right\} \) of \(n\times n\) matrices over a field \(F\), consider the set consisting of the \(m!\) products of the matrices in every possible order. If the matrices are nonsingular then any two products related by a cyclic permutation are similar, but in general these \((m-1)!\) similarity classes may all have different traces. When the matrices are singular the situation is more complicated, even for \(m=2\) [see \textit{H. Flanders}, Proc. Am. Math. Soc. 2, 871--874 (1951; Zbl 0044.00602)]. In the present paper the authors consider three situations: (SOI) all products are in the same similarity class; (EOI) all products have the same characteristic polynomial; and (TOI) all products have the same trace. The authors prove various properties of these classes, mainly for \(m=3\). A sample theorem is the following. If \(m=3\), \(n=2\) and \(char(F)\neq2\), then (TOI) holds if and only if \(A_{1},A_{2}\) and \(A_{3}\) are simultaneously symmetrizable or simultaneously upper triangularizable. The analogous result is false when \(m>3\).
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 5677359
zbMATH Keywords: permuted products
zbMATH Keywords: products of matrices
zbMATH Keywords: eigenvalues
zbMATH Keywords: traces
zbMATH Keywords: similarity classes
zbMATH Keywords: characteristic polynomial
MaRDI profile type: Publication
OpenAlex ID: W1991736581
cites work: Elementary Divisors of AB and BA
cites work: A note on sign-nonsingular matrices
cites work: Matrix Analysis
cites work: An extension of flanders theorem to several matrices
cites work: Generalization of Flanders' theorem to matrix triples.
cites work: Explicit Jordan form for certain block triangular matrices
cites work: The Relationship Between AB and BA
DOI: 10.1016/J.LAA.2009.04.014
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description: scientific article; zbMATH DE number 5677359