H. Rojo

An always nontrivial upper bound for Laplacian graph eigenvalues

Abstract Let G be a graph on vertex set V= v 1 ,v 2 ,…,v n . Let d i be the degree of v i , let N i be the set of neighbours of v i and let |S| be the number of vertices of S⊆V. In this note, we prove that max d i +d j −|N i ∩N j |:1⩽i is an upper bound for the largest eigenvalue of the Laplacian matrix of G. For any G, this bound does not exceed the order of G.

Bounds for sums of eigenvalues and applications

Let A be a matrix of order n × n with real spectrum λ1 ≥ λ2 ≥ ⋯ ≥ λn. Let 1 ≤ k ≤ n − 2. If λn or λ1 is known, then we find an upper bound (respectively, lower bound) for the sum of the k-largest (respectively, k-smallest) remaining eigenvalues of A. Then, we obtain a majorization vector for (λ1, λ2,…, λn−1) when λn is known and a majorization vector for (λ2, λ3,…, λn) when λ1 is known. We apply these results to the eigenvalues of the Laplacian matrix of a graph and, in particular, a sufficient condition for a graph to be connected is given. Also, we derive an upper bound for the coefficient of ergodicity of a nonnegative matrix with real spectrum.

New eigenvalue estimates for complex matrices

Abstract In a recent work [1], we have proved that all the eigenvalues λj of a complex matrix A of order n lie in a disk with center at tr A n and radius [ n−1 n (Σ k=1 n |λ k | 2 − | tr A | 2 n )] 1 2 . In this paper, we prove that the eigenvalues of A are contained in a rectangle with vertices on the boundary circle of the mentioned disk, improving in this way the result in [1]. We recall some inequalities which alloww us to bound the radius of the disk and the rectangle. The bounds can be computed without knowing the eigenvalues of A.

A Decreasing Sequence of Eigenvalue Localization Regions

Abstract We construct a decreasing sequence of rectangles (Rp) in such a way that all the eigenvalues of a complex matrix A are contained in each rectangle. When A is a matrix with real spectrum or a normal matrix, each Rp can be obtained without knowing the eigenvalues of A.

Constructing symmetric nonnegative matrices via the fast fourier transform

Abstract We derive an algorithm based on the fast Fourier transform to construct a real symmetric matrix S with eigenvalues λ 1 ≥ λ 2 ≥⋯≥ λ n , with eigenvector e = [1, 1, … ,1] T belonging to the eigenvalue λ 1 .We find simple conditions on the eigenvalues such that the algorithm constructs an irreducible matrix S = λ 1 E , where E is a symmetric doubly stochastic matrix.

Fast construction of a symmetric nonnegative matrix with a prescribed spectrum

Abstract In this paper, for a prescribed real spectrum, using properties of the circulant matrices and of the symmetric persymmetric matrices, we derive a fast and stable algorithm to construct a symmetric nonnegative matrix which realizes the spectrum. The algorithm is based on the fast Fourier transform.

Bounds for the spectral radius and the largest singular value

Abstract We derive an increasing sequence of lower bounds for the spectral radius of a matrix with real spectrum and progressively improved bounds for the largest singular value of a complex matrix. We also find estimates for the rank of normal matrices with real spectrum and for the rank of normal nonnegative matrices, including some sufficient condition for such matrices to be invertible.

Localization of eigenvalues in elliptic regions

Abstract We prove that the eigenvalues λ j of an n by n complex matrix A with its characteristic polynomial having real coefficients lie in the elliptic region defined by β 2 χ− tr A n 2 +α 2 y 2 ≤α 2 β 2 , where α n−1 n ∑ n=1 n ( Re λ k ) 2 − ( Re(te A )) 2 n 1 2 and β= n−1 n ∑ k=1 n ( Im λ k ) 2 1 2 This region is intersected with the strip |y|≤ 1 2 ∑ k=1 u Im λ k ) 2 1 2 to obtain an improved eigenvalue localization region. We also give bounds for the semiaxes, which can be computed without knowing the eigenvalues of A . When A has r n nonzero eigenvalues, we obtain a smaller elliptic region containing such nonzero eigenvalues.

Related bounds for the extreme eigenvalues

Abstract We show how to construct an increasing (decreasing) sequence of lower (upper) bounds for the smallest (largest) eigenvalue of a matrix with real spectrum by using the knowledge of the largest (smallest) eigenvalue. We apply these results to obtain a decreasing sequence of upper bounds for the largest singular value of a complex matrix. Finally, we show that the new bounds improve some previous results. Examples in which the bounds could be useful are given.

Polynomial basis functions for curved elements using hyperbolas

Abstract A conforming polynomial second order basis for the three sided two-dimensional finite elements with one curved side is constructed in such a way that the curved side is approximated by an arc of hyperbola. The basis is used to calculate approximate solutions of Laplaces equation over the unitary disk with Dirichlet boundary conditions. The basis has the property that it remains conforming when the curved side reverts to a straight line segment. The calculations of the typical integrals are made directly in the original domain of interest without the use of a non-linear transformation that is required in the high order transformation methods. Various tesselations of the problem domain were done and the numerical experiments show that the results are completely satisfactory for all the examples considered.