Copositive matrices, their dual, and the Recognition Problem

Abstract: Copositivity is a generalization of positive semidefiniteness. It has applications in economics, operations research, and statistics. An $n$-by-$n$ real matrix $A$ is copositive (CoP) if $x^TAx \ge 0$ for any nonnegative vector $x \ge 0$. The CoP matrices form a proper cone. A CoP matrix is ordinary if it can be written as the sum of a positive semidefinite (PSD) matrix and a symmetric nonnegative (sN) matrix. When $n < 5$, all copositive matrices are ordinary. However, recognition that a given CoP matrix is ordinary and the determination of an ordinary decomposition is an unresolved issue. Here, we make observations about CoP-preserving operations, make progress about the recognition problem, and discuss the relationship between the recognition problem and the PSD completion problem. We also mention the problem of copositive spectra and its relation to the symmetric nonnegative inverse eigenvalue problem.

combinatoricscomplex variablesfunctional analysisgeneral mathematicsgroup theoryK-theory and homologynumber theoryoperator algebrasprobabilityquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the discipline

GAG seminar

Series comments: Description: Non-specialized research seminar