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arXiv · 2410.18891

Uniqueness of size-2 positive semidefinite matrix factorizations

Kristen Dawson·Serkan Hoşten·Kaie Kubjas·Lilja Metsälampi
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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.

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Kristen Dawson, Serkan Hoşten, Kaie Kubjas, Lilja Metsälampi. 2024-10-24. Uniqueness of size-2 positive semidefinite matrix factorizations. https://arxiv.org/abs/2410.18891

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