Totally nonpositive completions on partial matrices

An n £ n real matrix is said to be totally no positive if every minor is no positive. In this paper, we are interested in totally no positive completion problems, that is, does A partial totally no positive matrix have a totally no positive matrix completion? This Problem has, in general, a negative answer. Therefore, we analyze the question: for which Labelled graphs G does every partial totally no positive matrix, whose associated graph is G, have a totally no positive completion? Here we study the mentioned problem when G Is a choral graph or an undirected cycle.Spanish DGI grant number BFM2001-0081-C03-02 and Generalitat Valenciana GRUPOS03/062Fundação para a Ciência e a Tecnologia (FCT

The doubly negative matrix completion problem

An $n\times n$ matrix over the field of real numbers is a doubly negative matrix if it is symmetric, negative definite and entry-wise negative. In this paper, we are interested in the doubly negative matrix completion problem, that is when does a partial matrix have a doubly negative matrix completion. In general, we cannot guarantee the existence of such a completion. In this paper, we prove that every partial doubly negative matrix whose associated graph is a $p$-chordal graph $G$ has a doubly negative matrix completion if and only if $p=1$. Furthermore, the question of completability of partial doubly negative matrices whose associated graphs are cycles is addressed.Spanish DGI - BFM2001-0081-C03-02.Fundação para a Ciência e a Tecnologia (FCT) – Programa Operacional “Ciência, Tecnologia, Inovação” (POCTI)

The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow

We provide several rigidity results for the Clifford torus in the class of compact self-shrinkers for Lagrangian mean curvature flow.Comment: 10 page

The symmetric N-matrix completion problem

An $n\times n$ matrix is called an $N$-matrix if all its principal minors are negative. In this paper, we are interested in the symmetric $N$-matrix completion problem, that is, when a partial symmetric $N$-matrix has a symmetric $N$-matrix completion. Here, we prove that a partial symmetric $N$-matrix has a symmetric $N$-matrix completion if the graph of its specified entries is chordal. Furthermore, if this graph is not chordal, then examples exist without symmetric $N$-matrix completions. Necessary and sufficient conditions for the existence of a symmetric $N$-matrix completion of a partial symmetric $N$-matrix whose associated graph is a cycle are given.Fundação para a Ciência e a Tecnologia (FCT) - Programa Operacional "Ciência, Tecnologia, Inovação" (POCTI). Spanish DGI - grant number BFM2001-0081-C03-02. Generalitat Valenciana - GRUPOS03/062

Eigenstructure of rank one updated matrices

[EN] The relationship among eigenvalues of a given square matrix A and the rank one updated matrix A+vkq⁎, where vk is an eigenvector of A associated with the eigenvalue λk and q is an arbitrary vector, was described by Brauer in 1952. In this work we study the relations between the Jordan structures of A and A+vkq⁎. More precisely, we analyze the generalized eigenvectors of the updated matrix in terms of the generalized eigenvectors of A, as well as the Jordan chains of the updated matrix. Further, we obtain similar results when we use a generalized eigenvector of A instead of the eigenvector vkSupported by the Spanish DGI grant MTM2013-43678-P.Bru García, R.; Cantó Colomina, R.; Urbano Salvador, AM. (2015). Eigenstructure of rank one updated matrices. Linear Algebra and its Applications. 485:372-391. https://doi.org/10.1016/j.laa.2015.07.036S37239148

Quasi-LDU factorization of nonsingular totally nonpositive matrices

Let A = (a(ij)) is an element of R-nxn be a nonsingular totally nonpositive matrix. In this paper we describe some properties of these matrices when a(11) = 0 and obtain a characterization in terms of the quasi-LDU factorization of A, where L is a block lower triangular matrix, D is a diagonal matrix and U is a unit upper triangular matrix. (c) 2012 Elsevier Inc. All rights reserved.The authors are very grateful to the referees for their helpful suggestions. This research was supported by the Spanish DGI Grant MTM2010-18228 and the Programa de Apoyo a la Investigacion y Desarrollo (PAID-06-10) of the Universitat Politecnica de Valencia.Cantó Colomina, R.; Ricarte Benedito, B.; Urbano Salvador, AM. (2013). Quasi-LDU factorization of nonsingular totally nonpositive matrices. Linear Algebra and its Applications. 439(4):836-851. https://doi.org/10.1016/j.laa.2012.06.010S836851439