Alberto Borobia

On the nonnegative eigenvalue problem

Abstract {1, λ1, …, λn} is the spectrum of a stochastic matrix of order n + 1 if and only if there exists some real matrix B of order n with spectrum {λ1, …, λn} and some n-simplex S ⊂ R n containing the origin such that BS ⊂ S. This result appears implicitly in Ciarlets work, and it provides us with a geometrical tool to obtain sufficient conditions for the nonnegative eigenvalue problem. We employ it here to generalize the sufficient conditions given by Kellogg.

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.

On nonnegative matrices similar to positive matrices

Abstract Let Nn denote the set of those (1, λ2, …, λn) ∈ C n such that there exists a nonnegative matrix with Perron root equal to one and spectrum {1, λ2, …, λn}. We prove that Nn is star-shaped with respect to (1, 0, …, 0) and that (1, λ 2 , …, λ n ) ∈ N n is on the boundary of Nn if and only if {1, λ2, …, λn} is not the spectrum of any positive matrix. As a consequence, attention is given to the problem of determining which nonnegative matrices are similar to positive ones. More precisely, we address the question of which pattern matrices P satisfy that any nonnegative matrix with pattern P is similar to a positive matrix. Some partial results are obtained (among them that any irreducible nonnegative matrix with a positive line is similar to a positive matrix), which allow us to give a complete solution to the case of 3-by-3 matrices.

MATRIX SCALING : A GEOMETRIC PROOF OF SINKHORN'S THEOREM

Abstract Matrix scaling problems have been extensively studied since Sinkhorn established in 1964 the following result: Any positive square matrix of order n is diagonally equivalent to a unique doubly stochastic matrix of order n, and the diagonal matrices which take part in the equivalence are unique up to scalar factors . We present a new elementary proof of the existence part of Sinkhorns theorem which is based on well-known geometrical interpretations of doubly stochastic matrices and left and right multiplication by diagonal matrices.

Three coefficients of a polynomial can determine its instability

Abstract We will prove that, in some cases, if we know only three coefficients of a polynomial with positive coefficients and without any restriction on the magnitude of its degree, we can conclude that the polynomial is unstable. Namely, if P(x)=∑ i=0 2n a i x 2n−i is a polynomial with positive coefficients and for some q∈{1,…,n−1} it is satisfied that a 2q n q )a 0 (n−q)/n a 2n q/n , then P(x) is unstable.

(0, ½, 1) matrices which are extreme points of the generalized transitive tournament polytope

Abstract Following Brualdi and Hwang, given a generalized transitive tournament (GTT) matrix T of order n , we consider the *-graph of T , that is, the undirected graph with vertices 1, 2, …, n in which there is an edge { i , j } between vertices i and j if and only if 0 ij . We characterize the *-graphs of the extreme GTT (0, ½, 1) matrices of order n . Using this characterization, we obtain for n = 6, 7 the complete list of extreme GTT (0, ½, 1) matrices of order n .

Graphs of vertices of the generalized transitive tournament polytope

Abstract A nonnegative matrix T = ( t ij ) i , j =1 n is a generalized transitive tournament matrix (GTT matrix) if t ii = 0, t ij = 1 − t ji for i ≠ j , and 1 ⩽ t ij + t jk + t ki ⩽ 2 for i , j , k pairwise distinct. The problem we are interested in is the characterization of the set of vertices of the polytope n of all GTT matrices of order n . In 1992, Brualdi and Hwang introduced the ∗ -graph associated to each T ∈ n . We characterize the comparability graphs of n vertices which are the ∗ -graphs of some vertex of n . As an application of the theoretical work we conclude that no comparability graph of at most 6 vertices and with at least one edge is the ∗ -graph of a vertex. In order to obtain the set of all vertices of 6 it only remains to analyse two noncomparability graphs.

Vertices of the generalized transitive tournament polytope

Abstract A nonnegative matrix T = (tij)i,j=tn is a generalized transitive tournament matrix (GTT matrix) if tjj = 0, tij = 1 −tji for i ≠ j, and 1 ⩽ tij + tjk + tki ⩽ 2 for i,j,k pairwise distinct. An approach to the problem of characterize the set of vertices of the polytope GTTn of all GTT matrices of order n was the introduction by Brualdi and Hwang of the ∗-graph associated to each T ϵ GTTn. We introduce a new graph which generalize the ∗-graph. The new graph will be employed to develop a computable criterion for determine whether any given GTT matrix of order n is or not a vertex of GTTn. A consequence of the criterion is that if T is a vertex of GTTn. with small number, r, of different entries then we have strong restrictions for the possible entries of T. Namely, if r ⩽ 6 then the set of entries of T is equal to 0, 1, 0, 1, 0, 1 2 ,1,0, 1 3 , 2 3 ,1, 0, 1 3 , 1 2 , 2 3 ,1, 0, 1 4 , 1 2 , 3 4 ,1, 0, 1 6 , 1 3 , 2 3 , 5 6 ,1, or 1 5 , 2 5 , 3 5 , 4 5 ,1.

Doubly stochastic matrices and dicycle covers and packings in eulerian digraphs

Abstract Mirsky (1963) raised the question of characterizing Ω 0 n , the convex hull of the nonidentity permutation matrices of order n, by a set of linear constraints. Cruse (1979) solved Mirskys problem by presenting an implicit description of those constraints. We associate an eulerian digraph with each doubly stochastic matrix, and then restate Cruses characterization of the polytope Ω 0 n in terms of dicycle covers of these digraphs. Brualdi and Hwang (1992) have shown, by using Cruses characterization and a result of Dridi (1980), an explicit set of linear inequalities that characterize Ω 0 n for n ⩽ 6. By using our characterization of Ω 0 n , we show that their result is valid if and only if n ⩽ 6. We show as well that if D is an eulerian digraph on n ⩽ 6 nodes, then there is always a minimum dicycle cover which is integral. We apply this last result and a result of Seymour (1994) to derive a min-max relation for eulerian digraphs on n ⩽ 6 nodes.

The real nonnegative inverse eigenvalue problem is NP-hard

Abstract A list of complex numbers is realizable if it is the spectrum of a nonnegative matrix. In 1949 Suleimanova posed the nonnegative inverse eigenvalue problem (NIEP): the problem of determining which lists of complex numbers are realizable. The version for reals of the NIEP (RNIEP) asks for realizable lists of real numbers. In the literature there are many sufficient conditions or criteria for lists of real numbers to be realizable. We will present an unified account of these criteria. Then we will see that the decision problem associated to the RNIEP is NP-hard and we will study the complexity for the decision problems associated to known criteria.