E. Marques de Sá

Imbedding conditions for λ-matrices

Abstract We say that A (λ) is λ-imbeddable in B (λ) whenever B (λ) is equivalent to a λ-matrix having A (λ) as a submatrix. In this paper we solve the problem of finding a necessary and sufficient condition for A (λ) to be λ-imbeddable in B (λ). The solution is given in terms of the invariant polynomials of A (λ) and B (λ). We also solve an analogous problem when A (λ) and B (λ) are required to be equivalent to regular λ-matrices. As a consequence we give a necessary and sufficient condition for the existence of a matrix B , over a field F , with prescribed similarity invariant polynomials and a prescribed principal submatrix A .

On the inertia of sums of Hermitian matrices

Abstract Given n×n Hermitian matrices, H1,…,Hp, a complete description is found for the possible inertias of the sum H1+⋯+Hp, in terms of the inertias of Hi, i=1,…,p. We also consider and solve other related problems.

The inertia of a Hermitian matrix having prescribed complementary principal submatrices

Abstract For i=1,2 let Hi be a given ni×ni Hermitian matrix. We characterize the set of inertias In H 1 X X ∗ H 2 :X is n 1 ×n 2 in terms of In(H1) and In(H2).

The inertia of hermitian matrices with a prescribed 2×2 block decomposition

Let denote given nonnegative integers, and consider the Hermitian matrices where each Hi =Hi* is ni × ni . We characterize the sets of inertias In each case the possible values of In(H)=(πν*) are characterized by a system of linear inequalities involving π,ν and the given integers.

The inertia of certain Hermitian skew-triangular block matrices

Abstract Let n1, n2, n3 be nonnegative integers. We consider Hermitian matrices H of the form H= H 11 H 12 H 13 H 21 H 22 H 23 H 31 H 32 H 33 where each Hij is ni × nj. We characterize the set of inertias {In(H): In(Hii) = (πi, vi, δi) and r1, i + 1 ⩽ rank H1, i + 1 ⩽ R1, i + 1 for i = 1, 2} in terms of π1, v1, δ1, π2, v2, δ2, n3, r12, r13, R12, R 12, and we discuss the implications of this characterization for the determination of the inertia of other types of Hermitian skew-triangular block matrices.

A convexity lemma on the interlacing inequalities for invariant factors

Abstract We give a simple proof of a discrete-convexity lemma to be used in the proof of sufficiency of the so-called interlacing inequalities for invariant factors . Two generalizations of the lemma are also proven.

Ranks of submatrices and the off-diagonal indices of a square matrix

Abstract In the first part of the paper we determine bounds for the ranks of certain submatrices of square matrices taken from a prescribed similarity class. Then we discuss the concept of off-diagonal indices (defined in the Introduction) which, very roughly speaking, measure, for each given integer s, how far we have to go off the main diagonal of a square matrix, to find an s×s nonzero minor. Some open problems are stated.

On the existence of matrices with constrained elements, rows and columns

Abstract Let the following be given: two n × m real matrices, E and F , such that F ⩽ E , three real n -rows, p, a and b , such that b ⩽ a , and three real m -columns, t, c and d , such that d ⩽ c . We give inequalities relating the given matrices and vectors, equivalent to the consistency of the system F ⩽ X ⩽ E , d ⩽ Xt ⩽ c , b ⩽ pX ⩽a , where X is an n × m unknown real matrix.