On the necessary and sufficient condition for a set of matrices to commute and some further linked results
type: Item
instance of: scholarly article
title: On the necessary and sufficient condition for a set of matrices to commute and some further linked results
zbMATH Open document ID: 1183.15014
DOI: 10.1155/2009/650970
published in: Mathematical Problems in Engineering
publication date: 2009-11-13T00:00:00Z
full work available at URL: https://eudml.org/doc/45875
review text: Summary: The author investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator \([A,B]\) for two matrices \(A\) and \(B\) equals zero if and only if a vector \(v(B)\) defined uniquely from the matrix \(B\) is in the null space of a well-structured matrix defined as the Kronecker sum \(A\oplus ( - A^{\ast })\), which is always rank defective. This result is extendable directly to any countable set of commuting matrices. Complementary results are derived concerning the commutators of certain matrices with functions of matrices \(f(A)\) which extend the well-known sufficiency-type commuting result \([A,f(A)]=0\).
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 15A16
zbMATH DE Number: 5632517
zbMATH Keywords: commutativity
zbMATH Keywords: commutator
zbMATH Keywords: Kronecker sum
zbMATH Keywords: functions of matrices
Wikidata QID: Q57675335
author: M. De la Sen
MaRDI profile type: Publication
OpenAlex ID: W2142638675
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description: scientific article; zbMATH DE number 5632517