Minimal zeros of copositive matrices

Let $A$ be an element of the copositive cone ${\cal C}_n$. A zero $u$ of $A$ is a nonzero nonnegative vector such that $u^TAu = 0$. The support of $u$ is the index set \mbox{supp}u \subset \{1,\dots,n\} corresponding to the positive entries of $u$. A zero $u$ of $A$ is called minimal if there does not exist another zero $v$ of $A$ such that its support \mbox{supp}v is a strict subset of \mbox{supp}u. We investigate the properties of minimal zeros of copositive matrices and their supports. Special attention is devoted to copositive matrices which are irreducible with respect to the cone $S_+(n)$ of positive semi-definite matrices, i.e., matrices which cannot be written as a sum of a copositive and a nonzero positive semi-definite matrix. We give a necessary and sufficient condition for irreducibility of a matrix $A$ with respect to $S_+(n)$ in terms of its minimal zeros. A similar condition is given for the irreducibility with respect to the cone ${\cal N}_n$ of entry-wise nonnegative matrices. For $n = 5$ matrices which are irreducible with respect to both $S_+(5)$ and ${\cal N}_5$ are extremal. For $n = 6$ a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided.Comment: Some conditions and proofs simplifie

Minimal zeros of copositive matrices

Let $A$ be an element of the copositive cone \copos{n}. A zero $u$ of $A$ is a nonzero nonnegative vector such that $u^TAu = 0$. The support of $u$ is the index set \Supp{u} \subset \{1,\dots,n\} corresponding to the positive entries of $u$. A zero $u$ of $A$ is called minimal if there does not exist another zero $v$ of $A$ such that its support \Supp{v} is a strict subset of \Supp{u}. We investigate the properties of minimal zeros of copositive matrices and their supports. Special attention is devoted to copositive matrices which are irreducible with respect to the cone $S_+(n)$ of positive semi-definite matrices, i.e., matrices which cannot be written as a sum of a copositive and a nonzero positive semi-definite matrix. We give a necessary and sufficient condition for irreducibility of a matrix $A$ with respect to $S_+(n)$ in terms of its minimal zeros. A similar condition is given for the irreducibility with respect to the cone \NNM{n} of entry-wise nonnegative matrices. For $n = 5$ matrices which are irreducible with respect to both $S_+(5)$ and \NNM{5} are extremal. For $n = 6$ a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided

Preservers of spectral radius, numerical radius, or spectral norm of the sum on nonnegative matrices

Let M(n)(+) be the set of entry wise nonnegative n x n matrices. Denote by r(A) the spectral radius (Perron root) of A is an element of M(n)(+). Characterization is obtained for maps f : M(n)(+) -\u3e M(n)(+) such that r(f (A) + f (B)) = r(A + B) for all A, B is an element of M(n)(+). In particular, it is shown that such a map has the form A bar right arrow S(-1) AS or A bar right arrow S(-1)A(tr)S for some S is an element of M(n)(+) with exactly one positive entry in each row and each column. Moreover, the same conclusion holds if the spectral radius is replaced by the spectrum or the peripheral spectrum. Similar results are obtained for maps on the set of n x n nonnegative symmetric matrices. Furthermore, the proofs are extended to obtain analogous results when spectral radius is replaced by the numerical range, or the spectral norm. In the case of the numerical radius, a full description of preservers of the sum is also obtained. but in this case it turns out that the standard forms do not describe all such preservers. (C) 2008 Elsevier Inc. All rights reserved

On complex power nonnegative matrices

Power nonnegative matrices are defined as complex matrices having at least one nonnegative integer power. We exploit the possibility of deriving a Perron Frobenius-like theory for these matrices, obtaining three main results and drawing several consequences. We study, in particular, the relationships with the set of matrices having eventually nonnegative powers, the inverse of M-type matrices and the set of matrices whose columns (rows) sum up to one

The NIEP

The nonnegative inverse eigenvalue problem (NIEP) asks which lists of $n$ complex numbers (counting multiplicity) occur as the eigenvalues of some $n$-by-$n$ entry-wise nonnegative matrix. The NIEP has a long history and is a known hard (perhaps the hardest in matrix analysis?) and sought after problem. Thus, there are many subproblems and relevant results in a variety of directions. We survey most work on the problem and its several variants, with an emphasis on recent results, and include 130 references. The survey is divided into: a) the single eigenvalue problems; b) necessary conditions; c) low dimensional results; d) sufficient conditions; e) appending 0's to achieve realizability; f) the graph NIEP's; g) Perron similarities; and h) the relevance of Jordan structure