On a Generalized Eigenvalue Problem for Nonsquare Pencils

In this paper a generalized eigenvalue problem for nonsquare pencils of the form $A-\lambda B$ with $A, B\in {\bf C}^{m\times n}$ and $m>n$, which was proposed recently by Boutry, Elad, Golub, and Milanfar [SIAM J. Matrix Anal. Appl., 27 (2006), pp. 582-601], is studied. An algebraic characterization for the distance between the pair $(A, B)$ and the pairs $(A_0, B_0)$ with the property that for the pair $(A_0, B_0)$ there exist $l$ distinct eigenpairs of the form $(A_0-\lambda_k B_0)\uv_k=0$, $k=1,\ldots, l$, is given, which implies that this distance can be obtained by solving an optimization problem over the compact set $\{ V_l: V_l\in {\bf C}^{n\times l}, V_l^HV_l=I\}$. Furthermore, the distance between a controllable descriptor system and uncontrollable ones is also considered, an algebraic characterization is obtained, and hence a well-known result on the distance between a controllable linear time-invariant system to uncontrollable ones is extended to the descriptor systems.