The Factor Width Rank of a Matrix

TL;DR

This paper introduces the concept of factor width rank of a matrix, exploring its properties, bounds, and relationships with graph theory, and analyzing how it behaves under various matrix operations.

Contribution

It defines factor width rank, establishes its properties, and connects it to graph clique covering, providing bounds and analyzing its behavior under matrix operations.

Findings

Factor width rank equals usual rank for banded and arrowhead matrices.

A tight connection exists between factor width rank and the k-clique covering number.

Factor width and rank can differ for certain matrices.

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.