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On eigenvalues of rectangular matrices

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Abstract

Given a (k+1)-tuple A,B 1, ..., B k of m×n matrices with m ≤ n, we call the set of all k-tuples of complex numbers {λ 1, ..., λ k} such that the linear combination A+λ 1 B 1+λ 2 B 2+ ... +λ k B k has rank smaller than m the eigenvalue locus of the latter pencil. Motivated primarily by applications to multiparameter generalizations of the Heine-Stieltjes spectral problem, we study a number of properties of the eigenvalue locus in the most important case k = n−m+1.

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Authors and Affiliations

  1. Department of Mathematics, Stockholm University, SE-106 91, Stockholm, Sweden

    Boris Shapiro

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824-1027, USA

    Michael Shapiro

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  1. Boris Shapiro
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  2. Michael Shapiro
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Correspondence to Boris Shapiro.

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Shapiro, B., Shapiro, M. On eigenvalues of rectangular matrices. Proc. Steklov Inst. Math. 267, 248–255 (2009). https://doi.org/10.1134/S0081543809040208

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  • DOI: https://doi.org/10.1134/S0081543809040208

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