The factor width rank of a matrix

Abstract:

A matrix is said to have factor width at most k if it can be written as a sum of positive semidefinite matrices that are non-zero only in a single k-by-k principal submatrix. We explore the “factor-width-k rank” of a matrix, which is the minimum number of rank-1 matrices that can be used in such a factor-width-at-most-k decomposition. We show that the factor width rank of a banded or arrowhead matrix equals its usual rank, but for other matrices they can differ. We also establish several bounds on the factor width rank of a matrix, including a tight connection between factor-width-k rank and the k-clique covering number of a graph, and we discuss how the factor width and factor width rank change when taking Hadamard products and Hadamard powers.

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  • N. Johnston, S. Moein, and S. Plosker. The factor width rank of a matrix. E-print: arXiv:2405.11556 [math.CO], 2024.
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