Related papers: Minimal zeros of copositive matrices

Let $n \geq 5$ and let $u^1,\dots,u^n$ be nonnegative real $n$-vectors such that the indices of their positive elements form the sets $\{1,2,\dots,n-2\},\{2,3,\dots,n-1\},\dots,\{n,1,\dots,n-3\}$, respectively. Here each index set is…

An $n\times n$ symmetric matrix $A$ is copositive if the quadratic form $x^TAx$ is nonnegative on the nonnegative orthant $\mathbb{R}^{n}_{\geq 0}$. The cone of copositive matrices contains the cone of matrices which are the sum of a…

The positive semidefinite rank of a nonnegative $(m\times n)$-matrix~$S$ is the minimum number~$q$ such that there exist positive semidefinite $(q\times q)$-matrices $A_1,\dots,A_m$, $B_1,\dots,B_n$ such that $S(k,\ell) = \mbox{tr}(A_k^*…

We give the criterion for the irreducibility, the Schur irreducibility and the indecomposability of the set of two $n\times n$ matrices $\Lambda_n$ and $A_n$ in terms of the subalgebra associated with the "support" of the matrix $A_n$,…

In this work we complement the description of the extreme rays of the $6 \times 6$ copositive cone with some topological structure. In a previous paper we decomposed the set of extreme elements of this cone into a disjoint union of pieces…

Let $A \in {\cal C}^n$ be an extremal copositive matrix with unit diagonal. Then the minimal zeros of $A$ all have supports of cardinality two if and only if the elements of $A$ are all from the set $\{-1,0,1\}$. Thus the extremal…

The paper is devoted to a study of the cone $\cop$ of copositive matrices. Based on the known from semi-infinite optimization concept of immobile indices, we define zero and minimal zero vectors of a subset of the cone $\cop$ and use them…

A nonnegative matrix $A$ is said to be {\it primitive} if for some positive integer $m$, entries in $A^m$ are positive, notationally represented as $A^m>0.$ The smallest such $m$ is called the {\it exponent} of $A$, denoted $exp(A).$ For…

The nonnegative rank of an entrywise nonnegative matrix A of size mxn is the smallest integer r such that A can be written as A=UV where U is mxr and V is rxn and U and V are both nonnegative. The nonnegative rank arises in different areas…

This paper investigates the uniqueness of a nonnegative vector solution and the uniqueness of a positive semidefinite matrix solution to underdetermined linear systems. A vector solution is the unique solution to an underdetermined linear…

We show that a matrix is a Hermitian positive semidefinite matrix whose nonzero entries have modulus 1 if and only if it similar to a direct sum of all $1's$ matrices and a 0 matrix via a unitary monomial similarity. In particular, the only…

Let $A$ be an $m \times n$ matrix with real entries. Given two proper cones $K_1$ and $K_2$ in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, we say that $A$ is nonnegative if $A(K_1) \subseteq K_2$. $A$ is said to be semipositive if…

This chapter investigates the cone of copositive matrices, with a focus on the design and analysis of conic inner approximations for it. These approximations are based on various sufficient conditions for matrix copositivity, relying on…

Topical maps are a nonlinear generalization of nonnegative matrices acting on the interior of the standard cone $\mathbb{R}^n_{\ge 0}$. Several analogues of irreducibility have been defined for topical maps, and all are sufficient to…

Many combinatorial matrices --- such as those of binomial coefficients, Stirling numbers of both kinds, and Lah numbers --- are known to be totally non-negative, meaning that all minors (determinants of square submatrices) are non-negative.…

Let $q$ be a prime power and $U$ the group of lower unitriangular matrices of order $n$ for some natural number $n$. We give a lower bound for the degrees of irreducible constituents of Andr\'{e}-Yan supercharacters and classify the…

Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory.…

An $n \times m$ non-negative matrix with row sum $m$ and column sum $n$ is called doubly stochastic. We answer the problem of finding doubly stochastic matrices of smallest posible support for every $1 <n \leq m$. Any matrix of minimum…

Let $\mathcal{S}_n$ be the set of all $n$-by-$n$ symmetric real matrices, and let $\mathcal{C}_n$ be the copositive cone, that is, the set of all matrices $a\in\mathcal{S}_n$ that fulfill the condition $u^\top a u\geqslant0$ for all…

A real symmetric n times n matrix is called copositive if the corresponding quadratic form is non-negative on the closed first orthant. If the matrix fails to be copositive there exists some non-negative certificate for which the quadratic…