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

    Johnson, C. R., Li, C. K., Rodman, L., Spitkovsky, I., & Pierce, S. (1992). Linear Maps Preserving Regional Eigenvalue Location. Linear and Multilinear Algebra, 32(3-4), 253-264. https://doi.org/10.1080/03081089208818168

    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, CR, Li, CK, Rodman, L, Spitkovsky, I & Pierce, S 1992, 'Linear Maps Preserving Regional Eigenvalue Location', Linear and Multilinear Algebra, vol. 32, no. 3-4, pp. 253-264. https://doi.org/10.1080/03081089208818168
    Johnson CR, Li CK, Rodman L, Spitkovsky I, Pierce S. Linear Maps Preserving Regional Eigenvalue Location. Linear and Multilinear Algebra. 1992 Oct 1;32(3-4):253-264. https://doi.org/10.1080/03081089208818168
    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.
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