A matrix $A$ is conjugate to $B$ if there exists an invertible matrix $P$ such that $$PAP^{-1}=B.$$ A matrix $X$ is completely positive if $X$ can be written as $$X=V^TV$$ for some non-negative matrix $V$ (not necessarily square). The cp-rank of $X$ is the smallest number of columns in such a matrix $V$.
I am interested in the relationship between the conjugacy classes and the cp-ranks of completely positive matrices. Specifically, I am wondering if $A$ is completely positive, what are some properties that one can determine for all the matrices $B$ that are conjugate to $A$ (i.e. exist $P$ such that $P^{-1}AP=B$)?
For example, are the cp-ranks of $A$ and $B$ always equal? If not, what are some bounds on their difference? Are there any conditions on $P$ that guarantee or prevent the equality of the cp-ranks?
Any references or insights on this topic would be appreciated. Thank you