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

AbstractFollowing Brualdi and Hwang, given a generalized transitive tournament (GTT) matrix T of ordern, we consider the *-graph ofT, that is, the undirected graph with vertices 1, 2, …,n in which there is an edge {i,j} between verticesi andj if and only if0 < tij < 1. 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 ordern

Three coefficients of a polynomial can determine its instability

AbstractWe 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=02naix2n−i is a polynomial with positive coefficients and for some q∈{1,…,n−1} it is satisfied that a2q<(nq)a0(n−q)/na2nq/n, then P(x) is unstable

A Geometric Proof of the Perron-Frobenius Theorem

Sin resume

A matrix completion problem over integral domains: the case with 2n-3 prescribed entries

AbstractLet Λ={λ1,…,λn}, n⩾2, be a given multiset of elements in an integral domain R and let P be a matrix of order n with at most 2n-3 prescribed entries that belong to R. Under the assumption that each row, each column and the diagonal of P have at least one unprescribed entry, we prove that P can be completed over R to obtain a matrix A with spectrum Λ. We describe an algorithm to construct A. This result is an extension to integral domains of a classical completion result by Herskowitz for fields

On the Consistency of the Matrix Equation X⊤ AX = B when B is Symmetric

We provide necessary and sufficient conditions for the matrix equation X⊤AX=B to be consistent when B is a symmetric matrix, for all matrices A with a few exceptions. The matrices A, B, and X (unknown) are matrices with complex entries. We first see that we can restrict ourselves to the case where A and B are given in canonical form for congruence and, then, we address the equation with A and B in such form. The characterization strongly depends on the canonical form for congruence of A. The problem we solve is equivalent to: given a complex bilinear form (represented by A) find the maximum dimension of a subspace such that the restriction of the bilinear form to this subspace is a symmetric non-degenerate bilinear form