  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.
