Linear Maps Preserving Regional Eigenvalue Location

Charles R. Johnson, Chi Kwong Li, Leiba Rodman, Ilya Spitkovsky, Stephen Pierce

Research output: Contribution to journalArticle

Abstract

Let M(n,C) be the vector space of n × n complex matrices and let G(r,s,t) be the set of all matrices in M(n,C) having r eigenvalues with positive real part, s eigenvalues with negative real part and t eigenvalues with zero real part. In particular, G(0,n,0) is the set of stable matrices. We investigate the set of linear operators on M(n,C) that map G(r,s,t) into itself. Such maps include, but are not always limited to similarities, transposition, and multiplication by a positive constant. The proof of our results depends on a characterization of nilpotent matrices in terms of matrices in a particular G(r,s,t), and an extension of a result about the existence of a matrix with prescribed eigenstructure and diagonal entries. Each of these results is of independent interest. Moreover, our characterization of nilpotent matrices is sufficiently general to allow us to determine the preservers of many other “inertia classes.”.

Original languageEnglish (US)
Pages (from-to)253-264
Number of pages12
JournalLinear and Multilinear Algebra
Volume32
Issue number3-4
DOIs
StatePublished - Oct 1 1992

ASJC Scopus subject areas

  • Algebra and Number Theory

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