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arXiv · 2405.11556

The Factor Width Rank of a Matrix

Nathaniel Johnston·Shirin Moein·Sarah Plosker
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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 $k \times k$ 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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Nathaniel Johnston, Shirin Moein, Sarah Plosker. 2025-04-02. The Factor Width Rank of a Matrix. https://doi.org/10.1016/j.laa.2025.03.016

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