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In this paper a generalized eigenvalue problem for nonsquare pencils of the form A - λB with A,B ϵ Cmxn 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 (A0,B0) with the property that for the pair (A0,B0) there exist l distinct eigenpairs of the form (A0-λkB0)v-k = 0, k = 1, … , l, is given, which implies that this distance can be obtained by solving an optimization problem over the compact set {Vl : Vl ϵ Cnxl, VH l VHlVl = 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. [ABSTRACT FROM AUTHOR]

The row by row decoupling problem (RRDP) for descriptor systems is considered using proportional state feedback and input transformation. Necessary and sufficient conditions for the solvability of the RRDP are provided. These solvability conditions can be readily verified. A constructive solution to the RRDP is given so that the desired feedback and input transformation matrices can be obtained by a numerically reliable procedure. [ABSTRACT FROM AUTHOR]

In this paper new necessary and sufficient conditions are given for the disturbance decoupling problem with or without stability and pole placement for linear time-invariant systems with direct feedthrough matrices, based on matrix pencil theory. All results are proved on the basis of a condensed form that can be computed using orthogonal matrix transformations, i.e., transformations that can be implemented in a numerically stable way. [ABSTRACT FROM AUTHOR]