Spectrally complete pairs of matrices
type: Item
instance of: scholarly article
title: Spectrally complete pairs of matrices
zbMATH Open document ID: 0652.15007
DOI: 10.1016/0024-3795(88)90191-7
author: S. H. Smith
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: A pair (A,B) of \(n\times n\) matrices over a field F is spectrally complete if for every n-tuple \((\lambda_ 1,...,\lambda_ n)\) of elements of F satisfying \(\Sigma \lambda_ i=tr A+tr B\) there is an A' similar to A such that \(A'+B\) has eigenvalues \(\lambda_ 1,...,\lambda_ n\). We denote by i(A) the number of invariant polynomials of A different from 1. The following theorem is the main result: Theorem. Let F be a field, and \(A,B\in F^{n\times n}\), with \(n>2\). Let \(f_ 1(x),...,f_ r(x)(g_ 1(x),...,g_ s(x))\) be the invariant polynomials of A(B) different from 1. (a) If \(i(A)+i(B)\leq n\) and at least one of the following conditions is satisfied: (1) \(n=2\), (2) there is at least one polynomial in the set \(\{f_ 1(x),...,f_ r(x),g_ 1(x),...,g_ s(x)\}\) which does not have degree two, then (A,B) is spectrally complete. (b) The converse of (a) is also true, if F has at least three elements.
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A04
zbMATH DE Number: 4063272
zbMATH Keywords: companion matrix
zbMATH Keywords: nonderogatory
zbMATH Keywords: eigenvalues
zbMATH Keywords: spectrally complete
zbMATH Keywords: invariant polynomials
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90191-7
OpenAlex ID: W2101782040
cites work: On the eigenvalues of the matrix
cites work: On the number of invariant polynomials of the matrix \(XAX^{-1}+B\)
cites work: Matrices with prescribed eigenvalues and principal submatrices
description: scientific article; zbMATH DE number 4063272