On Eigenvalues of Rectangular Matrices

Given a $(k+1)$-tuple $A,B_1,\dots,B_k$ of $m\times n$ matrices with $m\le n$, we call the set of all $k$-tuples of complex numbers $\{\lambda_1,\dots,\lambda_k\}$ such that the linear combination $A+\lambda_1B_1+\lambda_2B_2+\dots+\lambda_kB_k$ 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 $k=n-m+1$.