On eigenvalues of rectangular matrices
Boris Zalmanovich Shapiro, Michael Zalmanovich Shapiro · Proceedings of the Steklov Institute of Mathematics · 2009
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.