On eigenvalues of rectangular matrices

2007/11/22 by Julius Borcea, Boris Shapiro, Borcea, Julius +3 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematics and Applications #Matrix Theory and Algorithms #Primary 15A18 #Representation Theory (math.RT) #Secondary 15A22 #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.0711.3609

openalex publication_date 2007/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a (k+1)-tuple A, B₁,...,B_k of (m× n)-matrices with m≤ n we call the set of all k-tuples of complex numbers \\la₁,...,\la_k\ such that the linear combination A+\la₁B₁+\la₂B₂+...+\la_kB_k has rank smaller than m the \it eigenvalue locus of the latter pencil. Motivated primarily by applications to multi-parameter generalizations of the Heine-Stieltjes spectral problem, see \citeHe and \citeVol, we study a number of properties of the eigenvalue locus in the most important case k=n-m+1.

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