The Factor Width Rank of a Matrix

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 × 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.