Juan-Miguel Gracia

Sylvester Matrix Equation for Matrix Pencils

Abstract We study a homogeneous linear matrix equation system related to the strict equivalence of matrix pencils. We obtain the dimension of the vector space of its solutions in terms of the invariants of the strict equivalence. We give a characterization of the strict equivalence of matrix pencils by rank tests, and we extend Roths criterion for the corresponding nonhomogeneous system.

Nearest pair with more nonconstant invariant factors and pseudospectrum

Abstract Let (A,B)∈ C n×n × C n×m . Suppose that the number of nonconstant (i.e., ≠1 ) invariant factors of the polynomial matrix λ[I n ,0]−[A,B] is less than k. For all complex number λ denote by σ n−(k−1) (λ[I n ,0]−[A,B]) the greatest (n−(k−1)) th singular value of the matrix λ[I n ,0]−[A,B] . The minimum absolute value of the real function of complex variable λ↦σ n−(k−1) (λ[I n ,0]−[A,B]) gives the distance from (A,B) to the set of pairs with more or equal number of nonconstant invariant factors. When k=1 , this specializes in the formula of Eising for the distance from a controllable pair (A,B) to the nearest uncontrollable pair. The complex numbers λ lying in the sublevel set {λ∈ C | σ n (λ[ I n , 0 ]−[ A , B ])⩽e} of the function λ↦σ n (λ[I n ,0]−[A,B]), are the uncontrollable modes of all the pairs that are within an e tolerance of (A,B) . All the results of this paper are an immediate consequence of the Singular Value Decomposition of a matrix and of the interpretation of the singular values as the distances to the nearest matrices of lower ranks.

A characterization of feedback equivalence

This paper provides a new characterization of feedback equivalence that can be applied to controllable and noncontrollable matrix pairs

Geometric multiplicity margin for a submatrix

(A,B)

On similarities of class Cp and applications to matrix differential equations

. This result is based on a generalization of a theorem of Rosenbrock describing the closed-loop invariant polynomials that are attainable by applying state feedback to a given system

Derivatives of the diameter and the area of a connected component of the pseudospectrum

\dot x = Ax + Bu

Matrix pencil generated by a tensor product from two matrix pencils

.

Stirred tanks: a didactic tool

Let G be a square complex matrix with less than k nonconstant invariant factors. We find a complex matrix that gives an optimal approximation to G among all possible matrices that have more than or equal to k invariant factors, obtained by varying only the entries of a bottom right submatrix of G.

A Characterization of Feedback Equivalence Based on a Generalization of Rosenbrock's Theorem

Abstract This paper deals with similarities of class Cp, in particular reduction of class Cp to Jordan form and unitary triangularization of class Cp of a matrix function. The main result is applied to establish a method to lower the order of nonlinear matrix differential equations belonging to a broad class of equations. In particular, this method permits one to transform nonlinear equations of the first order into an algebraic equation joint to a linear differential equation of the first order. The method is applied to the matrix differential equation XX′ − X′X = X, and a particular solution is easily obtained.

Feedback Equivalence by Rank Tests

The paper concerns the relation between the following two quantities. (i) The Holder condition number of an eigenvalue λ of a square complex matrix. (ii) The rate of growth of the diameter and the area of the connected component of the e-pseusospectrum containing λ.