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Uniqueness of size-2 positive semidefinite matrix factorizations

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arxiv 2410.18891 v1 pith:NS2EBFGL submitted 2024-10-24 math.MG math.OC

Uniqueness of size-2 positive semidefinite matrix factorizations

classification math.MG math.OC
keywords rankfactorizationspositivesize-2matrixrigiditycharacterizeconditions
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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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