The CP-rank of a completely positive matrix 
is the least number of columns of a
matrix 
with nonnegative entries such that 
. Here, 
is the transpose of 
. The CP-rank is also called the factorization
index and is generally different from the ordinary matrix
rank. The CP-rank of the zero matrix is defined
to be 0.
Let 
be the largest CP-rank of an 
completely
positive matrix. Lavi (2026) proved a lower bound which, together with the known
upper bound, gives
 |
for 
, where 
for odd 
and 
for even 
. Consequently,
 |
