Susana Furtado

A generalization of Sylvester's law of inertia

Abstract We introduce the class of unitoid matrices as those that are diagonalizable by congruence; the nondegenerate canonical angles of a unitoid matrix A are the directions of the nonzero entries of a diagonal matrix congruent to A (which are shown to be unique). Sylvesters law states that two Hermitian matrices of the same size are congruent if and only if they have the same numbers of positive (respectively, negative) eigenvalues. Two unitoid matrices are congruent if and only if they have the same nondegenerate canonical angles.

On the characteristic polynomial of matrices with prescribed rows

Abstract Let A∈F n×n , B∈F n×t , where F is an arbitrary field. We describe the possible characteristic polynomials of [A B] , when some of its rows are prescribed and the other rows vary. The characteristic polynomial of [A B] is defined as the largest determinantal divisor (or the product of the invariant factors) of [xI n −A −B] . This result generalizes a previous theorem by H. Wimmer [Monasth. Math. 78 (1974) 256–263], which studies the same problem when t=0 .

Embedding a regular subpencil into a general linear pencil

Abstract We study the possible strictly equivalence classes of a pencil when a regular subpencil is prescribed. We also study the possible invariant polynomials and the possible characteristic polynomials of A+BY+XC+XDY when X and Y vary.

Spectral variation under congruence

In this paper we study the possible spectra among matrices congruent to a given AεMn (C). It is important to distinguish singular A from nonsingular and, among non-singular matrices, to distinguish where 0 lies relative to the field of values of A.

Products of matrices with prescribed spectra and ranks

This paper studies the possibility of writing a given square matrix as the product of two matrices with prescribed spectra and ranks. It extends some previously known results.

Linearizations of Hermitian Matrix Polynomials Preserving the Sign Characteristic

The development of strong linearizations preserving whatever structure a matrix polynomial might possess has been a very active area of research in the last years, since such linearizations are the starting point of numerical algorithms for computing eigenvalues of structured matrix polynomials with the properties imposed by the considered structure. In this context, Hermitian matrix polynomials are one of the most important classes of matrix polynomials arising in applications and their real eigenvalues are of great interest. The sign characteristic is a set of signs attached to these real eigenvalues which is crucial for determining the behavior of systems described by Hermitian matrix polynomials and, therefore, it is desirable to develop linearizations that preserve the sign characteristic of these polynomials, but, at present, only one such linearization is known. In this paper, we present a complete characterization of all the Hermitian strong linearizations that preserve the sign characteristic of ...

Spectral variation under congruence for a nonsingular matrix with 0 on the boundary of its field of values

Abstract In this paper we describe the possible spectra among matrices congruent to a given nonsingular matrix A∈M n ( C ) such that 0 is a boundary point of its field of values. In the process we give a reducible form under congruence for A . If A is either singular or such that 0 is not a boundary point of its field of values, the possible spectra among matrices congruent to A were studied in [Linear and Multilinear Algebra 49 (2001) 243].

An algorithm for constructing a pseudo‐Jacobi matrix from given spectral data

SUMMARY The main purpose of this paper is the extension of the classical spectral direct and inverse analysis of Jacobi matrices for the non-self-adjoint setting. Matrices of this class appear in the context of non-Hermitian quantum mechanics. The reconstruction of a pseudo-Jacobi matrix from its spectrum and the spectra of two complementary principal matrices is investigated in the context of indefinite inner product spaces. An existence and uniqueness theorem is given, and a strikingly simple algorithm, based on the Euclidean division algorithm, to reconstruct the matrix from the spectral data is presented. A result of Friedland and Melkman stating a necessary and sufficient condition for a real sequence to be the spectrum of a non-negative Jacobi matrix is revisited and generalized. Namely, it is shown that a suitable set of prescribed eigenvalues defines a unique non-negative pseudo-Jacobi matrix, which is J-Hermitian for a fixed J. Copyright

Products of Real Matrices with Prescribed Characteristic Polynomials

Let A be a matrix with entries in the field of real numbers. In this paper we give necessary and sufficient conditions for the existence of real matrices B and C, with prescribed characteristic polynomials, such that A=BC.

On the characteristic polynomial of matrices with prescribed columns and the stabilization and observability of linear systems

Let A 2 F , B 2 F , where F is an arbitrary eld. In this paper, the possible characteristic polynomials of [A B ], when some of its columns are prescribed and the other columns vary, are described. The characteristic polynomial of [A B ] is de ned as the largest determinantal divisor (or the product of the invariant factors) of [xIn A B ]. This result generalizes a previous theorem by H. Wimmer which studies the same problem when t = 0. As a consequence, it is extended to arbitrary elds a result, already proved for in nite elds, that describes all the possible characteristic polynomials of a square matrix when an arbitrary submatrix is xed and the other entries vary. Finally, applications to the stabilization and observability of linear systems by state feedback are studied. AMS subject classi cations. 15A18, 93B60