Nonnegative rank of a matrix with one negative eigenvalue

TL;DR

This paper demonstrates that symmetric matrices of rank three with a single negative eigenvalue can have arbitrarily large nonnegative rank, challenging assumptions about the relationship between eigenvalues and nonnegative rank.

Contribution

It establishes a surprising link between eigenvalue signatures and nonnegative rank, showing that a single negative eigenvalue does not bound the nonnegative rank.

Findings

Rank-three symmetric matrices with one negative eigenvalue can have arbitrarily large nonnegative rank.

The result challenges previous beliefs about eigenvalues constraining nonnegative rank.

The study provides new insights into the structure of nonnegative matrices with specific spectral properties.

Abstract

We show that a rank-three symmetric matrix with exactly one negative eigenvalue can have arbitrarily large nonnegative rank.