Ion Zaballa

Matrices with prescribed rows and invariant factors

Abstract In this paper, we solve the problem of the existence of an n × n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed. The solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials.

Interlacing inequalities and control theory

Abstract Given a p by p matrix A , we solve the problem of the existence of a p by q matrix B such that ( A , B ) has prescribed controllability indices and [ λI p − A ,− B ] has prescribed invariant polynomials. The solution of this problem, together with an earlier theorem of the authors, is used to provide a new proof of the Sa-Thompson interlacing inequalities for invariant polynomials.

Column completion of a pair of matrices

Given a pair of matrices (A B) it is well known that its invariant factors and its controllability indices form a complete set of invariants for the Γ-equivalence [11] or block similarity [5]. How do they vary by adding columns to B? This problem was solved in [12] when B = 0; here we give a complete answer for this question.

Controllability and Hermite indices of matrix pairs

Given a pair of matrices ( A, B ), we can associate to each nice basis of the controllability subspace of ( A, B ) a set of integers to be called the indices of the basis. Each set provides partial information about the similarity class of A . The relationship between several sets of indices as well as the constraints that they impose on the invariant factors of A are investigated. Special attention is paid to the controllability and Hermite indices of a given pair ( A, B ).

Assigning the Kronecker invariants of a matrix pencil by row or column completions

Abstract The challenge consists in describing the relationships between the Kronecker invariants of a matrix pencil and one of its subpencils. For a given subpencil, an algorithm for constructing a matrix pencil with prescribed Kronecker invariants should also be proposed.

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.

Matrices with prescribed invariant factors and off-diagonal submatrices

We solve the problem of the existence of a matrix with prescribed invariant factors and an off-diagonal submatrix (i.e., a submatrix without entries from the main diagonal and such that the number of rows plus the number of columns is the size of the given matrix). We will use some ideas from Control theory.

Interlacing and majorization in invariant factor assignment problems

This paper shows some connections between pole assignment in control theory and assignment of invariant factors on matrices with some prescribed submatrices. The majorization in the Hardy-Littlewood-Polya sense and the interlacing inequalities for invariant factors play a fundamental role in the solution of some problems common to both theories.

Existence of matrices with prescribed entries

Abstract It is proved that, apart from for some exceptional cases, there always exists an n × n nonderogatory matrix over an arbitrary field with n prescribed entries and prescribed characteristic polynomial.

Triangularizing Quadratic Matrix Polynomials

We show that any regular quadratic matrix polynomial can be reduced to an upper triangular quadratic matrix polynomial over the complex numbers preserving the finite and infinite elementary divisors. We characterize the real quadratic matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable with diagonal blocks of sizes