Geodesic
On Eigenvalues of Rectangular Matrices
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Singularities and applications, Tome 267 (2009), pp. 258-265.

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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 $k=n-m+1$. 
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B. Shapiro; M. Shapiro. On Eigenvalues of Rectangular Matrices. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Singularities and applications, Tome 267 (2009), pp. 258-265. http://geodesic.mathdoc.fr/item/TM_2009_267_a19/

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