The Factor Width Rank of a Matrix
| Autor: | Johnston, Nathaniel, Moein, Shirin, Plosker, Sarah |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 23 pages |
| Databáze: | arXiv |
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