C.M. da Fonseca

Explicit inverses of some tridiagonal matrices

Abstract We give explicit inverses of tridiagonal 2-Toeplitz and 3-Toeplitz matrices which generalize some well-known results concerning the inverse of a tridiagonal Toeplitz matrix.

THE INERTIA OF CERTAIN HERMITIAN BLOCK MATRICES

Abstract We characterize sets of inertias of some partitioned Hermitian matrices by a system of inequalities involving the orders of the blocks, the inertias of the diagonal blocks, and the ranks of the nondiagonal blocks. The main result generalizes some well-known characterizations of Sa and Cain and others.

Fibonacci numbers, alternating parity sequences and faces of the tridiagonal Birkhoff polytope

We determine the number of alternating parity sequences that are subsequences of an increasing m-tuple of integers. For this and other related counting problems we find formulas that are combinations of Fibonacci numbers. These results are applied to determine, among other things, the number of vertices of any face of the polytope of tridiagonal doubly stochastic matrices.

On the location of the eigenvalues of Jacobi matrices

Using some well known concepts on orthogonal polynomials, some recent results on the location of eigenvalues of tridiagonal matrices of very large order are extended. A significant number of important papers are unified.

The inertia of Hermitian block matrices with zero main diagonal

Abstract Let n 1 ,n 2 ,n 3 be nonnegative integers. We consider partitioned Hermitian matrices of the form H= 0 X 12 X 13 [3pt]X 12 * 0 X 23 [5pt]X 13 * X 23 * 0 , where each X ij is n i ×n j and we characterize the set of the inertias In (H) | r ij ⩽ rank X ij ⩽R ij for 1⩽i in terms of r ij , R ij and the block orders.

The inverse eigenvalue problem for Hermitian matrices whose graphs are cycles

In 1979, Ferguson characterized the periodic Jacobi matrices with given eigenvalues and showed how to use the Lanzcos Algorithm to construct each such matrix. This article provides general characterizations and constructions for the complex analogue of periodic Jacobi matrices. As a consequence of the main procedure, we prove that the multiplicity of an eigenvalue of a periodic Jacobi matrix is at most 2.

Nonsingular acyclic matrices with full number of P-vertices

We classify the trees for which there is a nonsingular matrix where each vertex is a P-vertex. In particular, we show that such trees have an even number of vertices. Both results provide answers to questions proposed by Kim and Shader. In the end, related classifications on nonsingular trees with the size of a P-set bounded are addressed.

The maximum number of P-vertices of some nonsingular double star matrices

Abstract In this short note, we construct a nonsingular matrix A whose graph is a double star of order n ⩾ 4 with n − 2 P-vertices. This example leads to a positive answer, for n ⩾ 6 , to a last open question proposed recently by Kim and Shader regarding the trees for which each nonsingular matrix has at most n − 2 P-vertices.

A note on the multiplicities of the eigenvalues of a graph

Let A(G) be a Hermitian matrix whose graph is a given graph G. From the interlacing theorem, it is known that , where is the multiplicity of the eigenvalue θ of A(G). In this note we improve this inequality for some paths with more than one vertex.Let A(G) be a Hermitian matrix whose graph is a given graph G. From the interlacing theorem, it is known that , where is the multiplicity of the eigenvalue θ of A(G). In this note we improve this inequality for some paths with more than one vertex.

Path polynomials of a circuit: a constructive approach

Let Pk denote the polynomial of the path on k vertices. We describe completely the matrix Pk (Cn ), where Cn is the circuit on n vertices, using some important concepts of theory of circulant matrices. We also consider Q k , the polynomial of the circuit on kvertices. Using orthogonal polynomials we present constructive proofs of some results obtained recently by Bapat and Lai, Beezer and Ronghua.