Juan M. Gracia

Perturbation of linear control systems

Abstract We study the variation of the controllability indices and the Jordan structure of a pair of matrices ( A , B ), under small perturbations.

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.

Stability of invariant subspaces of regular matrix pencils

Abstract We characterize the stable invariant (or deflating) subspaces of a regular matrix pencil.

Safety neighbournoods for the invariants of the matrix similarity

We give safety neighbourhoods for the necessary conditions in the change of the Jordan canonical form of a matrix under small perturbations. We also obtain the minimum distance from an n × n complex matrix which has less than k nonconstant invariant factors (2≤ k≤ n) to the set of matrices which have more or equal to k. When k= 2, we get in particular the distance from a nonderogatory matrix to the set of derogatory matrices.

Local behavior of Sylvester matrix equations related to block similarity

Topological properties of a matrix equation of Sylvester type are considered. This equation is related to the block similarity between rectangular matrices, and the passage matrices in this equivalence relation are solutions. A local criterion for that similarity is exposed. The points of continuity of the map that associates with the coefficient matrices the solution space of that equation are determined.

Periodic solutions of linear differential systems with constant coefficients

In this paper necessary and sufficient conditions are established, in terms of spectral properties of a complex n x nmatrix A, which warrants periodicity to the exponential matrix eAt and to all solutions of the linear differential system x’ = Ax,with precisions about the minimum positive period.

Dimension of the solution spaces of the matrix equations [A, [A, X]] = 0 and [A, [A, [A, X]]]= 0

In this paper, a formula for the dimension of spaces of solutions of these equations in terms of Segre characteristic of matrix A is given.

Global analytic block similarity to a Brunoovsky form

Abstract Necessary and sufficient conditions for reducing a pair of analytic matrix functions ( A ( z ), B ( z )) to a brunovsky canonical form by means of analytic passage matrices are given.

In variance under translations

In this paper, the functions which are solutions of constant coefficient homogeneous linear differential (or difference) equations are characterized in terms of the vector space spanned by the set of their translates. For this, a result about the differentiable character of the continuous solutions of Polyas functional equation is used.

On Pólya's functional equation

Let Y be a continuous matrix valued function defined on R, verifying the functional equation Y(s +t) = Y(s) Y(t) for all real s, t. This paper corrects certain mistakes found in an earlier elementary proof that Y is differentiable at 0. Also, reference is made to some relations of this equation with one‐parameter groups, with Chapman—Kolmogorov equation in Markov processes, with combinatorics, and with a characterization of the solutions of the constant coefficient homogeneous linear differential equations.