Uniqueness of size-2 positive semidefinite matrix factorizations

TL;DR

This paper characterizes when certain size-2 positive semidefinite factorizations of rank-3 matrices are unique, using rigidity theory to connect infinitesimal rigidity to global rigidity and boundary conditions.

Contribution

It introduces a characterization of uniqueness for size-2 psd factorizations of rank-3 matrices using rigidity concepts and boundary analysis.

Findings

Characterization of 1- and 2-infinitesimal rigidity for size-2 psd factorizations

Connection between infinitesimal rigidity and global rigidity for uniqueness

Necessary boundary conditions for matrices with given rank and psd rank

Abstract

We characterize when a size-2 positive semidefinite (psd) factorization of a positive matrix of rank 3 and psd rank 2 is unique. The characterization is obtained using tools from rigidity theory. In the first step, we define s-infinitesimally rigid psd factorizations and characterize 1- and 2-infinitesimally rigid size-2 psd factorizations. In the second step, we connect 1- and 2-infinitesimal rigidity of size-2 psd factorizations to uniqueness via global rigidity. We also prove necessary conditions on a positive matrix of rank 3 and psd rank 2 to be on the topological boundary of all nonnegative matrices with the same rank conditions.