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 is an element of C-mxn 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) (v) under bar (k) = 0, k = 1,..., l, is given, which implies that this distance can be obtained by solving an optimization problem over the compact set {V-l : V-l is an element of C-nxl, V-l(H) V-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.