M. Asunción Beitia

A linear matrix equation: a criterion for block similarity

In this paper we study a homogeneous linear matrix equation related to the block similarity of rectangular matrices. We obtain the dimension of the vector space of its solutions and we describe these solutions. We give a characterization of the block similarity by rank tests. We extend Roths criterion to the corresponding non homogeneous equation.

The change of feedback invariants under column perturbations: particular cases

We study the variation of the feedback invariants of a complex rectangular n × (n + m) matrix when we make small additive perturbations to the elements of the last m columns. First of all, we obtain necessary conditions for the feedback invariants of all the matrices obtained by means of sufficiently small perturbations. Conversely, we prove that these conditions are also sufficient to find a matrix, as close as desired to the fixed matrix, with prescribed feedback invariants, in some particular cases: when the rectangular matrix is completely controllable, when the rectangular matrix is completely uncontrollable and when m = 1.