On eigenvalues of rectangular matrices

TL;DR

This paper investigates the properties of eigenvalue loci associated with linear combinations of rectangular matrices, motivated by multi-parameter spectral problems, focusing on the case where the number of parameters matches the difference in matrix dimensions.

Contribution

It introduces the concept of eigenvalue loci for matrix pencils and analyzes their properties in the context of multi-parameter generalizations of spectral problems.

Findings

Characterization of eigenvalue loci for specific matrix configurations

Connections to multi-parameter spectral theory

Insights into rank conditions of matrix linear combinations

Abstract

Given a $(k+1)$-tuple $A, B_1,...,B_k$ of $(m\times n)$-matrices with $m\le n$ we call the set of all $k$-tuples of complex numbers $\{\la_1,...,\la_k\}$ such that the linear combination $A+\la_1B_1+\la_2B_2+...+\la_kB_k$ 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 \cite{He} and \cite{Vol}, we study a number of properties of the eigenvalue locus in the most important case $k=n-m+1$.