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

Fingerprint

Linear map
Eigenvalue
Nilpotent Matrix
Preserver
Transposition
Inertia
Linear Operator
Vector space
Multiplication
Zero

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Linear Maps Preserving Regional Eigenvalue Location. / Johnson, Charles R.; Li, Chi Kwong; Rodman, Leiba; Spitkovsky, Ilya; Pierce, Stephen.

In: Linear and Multilinear Algebra, Vol. 32, No. 3-4, 01.10.1992, p. 253-264.

Research output: Contribution to journalArticle

Johnson, Charles R. ; Li, Chi Kwong ; Rodman, Leiba ; Spitkovsky, Ilya ; Pierce, Stephen. / Linear Maps Preserving Regional Eigenvalue Location. In: Linear and Multilinear Algebra. 1992 ; Vol. 32, No. 3-4. pp. 253-264.
@article{6caef652ccc647ed9d36af95313ee6b7,
title = "Linear Maps Preserving Regional Eigenvalue Location",
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.”.",
author = "Johnson, {Charles R.} and Li, {Chi Kwong} and Leiba Rodman and Ilya Spitkovsky and Stephen Pierce",
year = "1992",
month = "10",
day = "1",
doi = "10.1080/03081089208818168",
language = "English (US)",
volume = "32",
pages = "253--264",
journal = "Linear and Multilinear Algebra",
issn = "0308-1087",
publisher = "Taylor and Francis Ltd.",
number = "3-4",

}

TY - JOUR

T1 - Linear Maps Preserving Regional Eigenvalue Location

AU - Johnson, Charles R.

AU - Li, Chi Kwong

AU - Rodman, Leiba

AU - Spitkovsky, Ilya

AU - Pierce, Stephen

PY - 1992/10/1

Y1 - 1992/10/1

N2 - 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.”.

AB - 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.”.

UR - http://www.scopus.com/inward/record.url?scp=84963333603&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84963333603&partnerID=8YFLogxK

U2 - 10.1080/03081089208818168

DO - 10.1080/03081089208818168

M3 - Article

AN - SCOPUS:84963333603

VL - 32

SP - 253

EP - 264

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

IS - 3-4

ER -