M. Isabel García-Planas

Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils

Abstract For a family of linear operators A( λ → ):U→U over C that smoothly depend on parameters λ → =(λ 1 ,…,λ k ) , V.I. Arnold obtained the simplest normal form of their matrices relative to a smoothly depending on λ → change of a basis in U . We solve the same problem for a family of linear operators A( λ → ):U→U over R , for a family of pairs of linear mappings A( λ → ):U→V, B( λ → ):U→V over C and R , and for a family of pairs of counter linear mappings A( λ → ):U→V, B( λ → ):V→U over C and R .

An alternative system of structural invariants of quadruples of matrices

Abstract The equivalence relation between time-invariant linear dynamical systems deduced from the action of the group consisting of all basis changes, state feedback and output injection transformations is considered. An alternative complete system of structural invariants is presented. Discrete invariants in this system are the ranks of suitable matrices, generalizing the numerical invariants obtained for pairs of matrices under block-similarity. The continuous invariants are characterized in terms of the rank of a matrix.

Generic families of matrix pencils and their bifurcation diagrams

Abstract V.I. Arnold [Russian Math. Surveys 26 (2) (1971) 29] constructed smooth generic families of matrices with respect to similarity transformations depending smoothly on the entries of matrices and got bifurcation diagrams of such families with a small number of parameters. We extend these results to pencils of matrices.

Associating matrix pencils to generalized linear multivariable systems

We consider quadruples of matrices (E,A,B,C) representing generalized linear multivariable systems Ex(t)=Ax(t)+Bu(t),y(t)=Cx(t), with E, A square matrices and B, C rectangular matrices. We characterize equivalent quadruples, by associating matrix pencils to them, with respect to the equivalence relation corresponding to standard transformations: basis changes (for the state, control and output spaces), state feedback, derivative feedback and output injection. Equivalent quadruples are those whose associated matrix pencils are “simultaneously equivalent”.

Families of Convolutional Codes over Finite Fields: A Survey

The goal of this work is to give explicit interconnections between control theory and coding. It is well-known the existence of a closed relation between linear systems over finite fields and convolutional codes that allow to understand some properties of convolutional codes and to construct them. The connection between convolutional codes and linear systems permit to consider control as well as analyze observability of convolutional codes under linear systems point of view.

Isometries of the hamming space and equivalence relations of linear codes over a finite field

Detection and error capabilities are preserved when applying to a linear code an isomorphism which preserves Hamming distance. We study here two such isomorphisms: permutation isometries and monomial isometries.