Unitary similarity of matrices with quadratic minimal polynomials
type: Item
instance of: scholarly article
title: Unitary similarity of matrices with quadratic minimal polynomials
zbMATH Open document ID: 1002.15012
DOI: 10.1016/S0024-3795(01)00606-1
author: Alan George
author: Kh. D. Ikramov
published in: Linear Algebra and its Applications
publication date: 2002-08-28T00:00:00Z
review text: The fact that given complex \(n\times n\) matrices \(A\) and \(B\) are (or are not) unitarily similar can be verified with the help of the Specht-Pearcy criterion [cf. \textit{C. Pearcy}, Pacific J. Math. 12, 1405-1416 (1962; Zbl 0144.37801)]. Its application involves a huge amount of computational work. For some matrices \(A,B\), most of this work is redundant. The paper identifies a class of matrices where unitary similarity among its members can be economically verified. These are matrices with quadratic minimal polynomials. If \(A\) and \(B\) are matrices of this kind, then, to be unitarily similar, they need to have the same eigenvalues and the same singular values, which can be verified by comparing two \(2n\)-tuples of traces. Two widely known subclasses of matrices with quadratic minimal polynomials are projectors and involutions. For these subclasses, the authors give yet another derivation of the unitary similarity criterion above, based on the canonical form for oblique projectors found by \textit{D. Ž. Doković} [Aequationes Math. 42, No. 2/3, 220-224 (1991; Zbl 0749.15005)].
reviewed by: Rodica Covaci
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 1790240
zbMATH Keywords: unitary similarity
zbMATH Keywords: eigenvalues
zbMATH Keywords: singular values
zbMATH Keywords: traces
zbMATH Keywords: quadratic minimal polynomials
zbMATH Keywords: canonical form
MaRDI profile type: Publication
cites work: Matrix Analysis
cites work: A complete set of unitary invariants for operators generating finite \(W^ *\)-algebras of type I
cites work: Unitary similarity of projectors
description: scientific article; zbMATH DE number 1790240