Uniqueness of size-2 positive semidefinite matrix factorizations
Metric Geometry
2024-10-25 v1 Optimization and Control
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.
Cite
@article{arxiv.2410.18891,
title = {Uniqueness of size-2 positive semidefinite matrix factorizations},
author = {Kristen Dawson and Serkan Hoşten and Kaie Kubjas and Lilja Metsälampi},
journal= {arXiv preprint arXiv:2410.18891},
year = {2024}
}
Comments
29 pages