Ricardo L. Soto

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.

Existence and construction of nonnegative matrices with prescribed spectrum

Abstract We consider the following inverse spectrum problem for nonnegative matrices: given a set of real numbers σ={λ1,λ2,…,λn}, find necessary and sufficient conditions for the existence of an n×n nonnegative matrix A with spectrum σ. In particular, by the use of a relevant theorem of Brauer we obtain new simple sufficient conditions for the problem to have a solution. Moreover, we can always construct a solution matrix, which is nonnegative generalized stochastic.

Nonnegative matrices with prescribed elementary divisors

The inverse elementary divisor problem for nonnegative matrices asks for necessary and su‐cient conditions for the existence of a nonnegative matrix with prescribed elementary divisors. In this work we introduce some perturbation results, which allow us to construct certain nonnegative matrices with arbitrarily prescribed elementary divisors.

Symmetric nonnegative realization of spectra

A perturbation result, due to R. Rado and presented by H. Perfect in 1955, shows how to modify r eigenvalues of a matrix of order n, r ≤ n, via a perturbation of rank r, without changing any of the n − r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank r = 1. Both results have been exploited in connection with the nonnegative inverse eigenvalue problem. In this paper a symmetric version of Rado’s extension is given, which allows us to obtain a new, more general, sufficient condition for the existence of symmetric nonnegative matrices with prescribed spectrum.

Existence and construction of nonnegative matrices with complex spectrum

Abstract The following inverse spectrum problem for nonnegative matrices is considered: given a set of complex numbers σ ={ λ 1 , λ 2 ,…, λ n }, find necessary and sufficient conditions for the existence of an n × n nonnegative matrix A with spectrum σ . Our work is motivated by a relevant theoretical result of Guo Wuwen [Linear Algebra Appl. 266 (1997) 261, Theorem 2.1]: there exists a real parameter λ 0 ⩾max 2⩽ j ⩽ n | λ j | such that the problem has a solution if and only if λ 1 ⩾ λ 0 . In particular, we discuss how to compute λ 0 and the solution matrix A for certain class of matrices. A sufficient condition for the problem to have a solution is also derived.

An inverse eigenvalue problem for symmetrical tridiagonal matrices

We consider the following inverse eigenvalue problem: to construct a symmetrical tridiagonal matrix from the minimal and maximal eigenvalues of all its leading principal submatrices. We give a necessary and sufficient condition for the existence of such a matrix and for the existence of a nonnegative symmetrical tridiagonal matrix. Our results are constructive, in the sense that they generate an algorithmic procedure to construct the matrix.

A family of realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem

SUMMARY A new realizability criterion for the real nonnegative inverse eigenvalue problemis introduced. This criterion is a nontrivial extension of a powerful previous sufficient condition, based on negativity compensation. If the criterion is satisfied, then we can always construct a realizing matrix. It is also proved that this new criterion is easily adaptable to be sufficient for the construction of a symmetric nonnegative matrixwith given spectrum. In a natural way, the criterion extends to a family of sufficient conditions for the problem to have a solution. Copyright

POSITIVE MATRICES WITH PRESCRIBED SINGULAR VALUES

We consider the problem of constructing positive matrices with prescribed singular values. In particular, we show how to construct an m × n positive matrix, m = n, with prescribed singular values s1 = s2 = · · · = sn.

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.