Preservers of the cp-rank
Naomi Shaked-Monderer · Linear and Multilinear Algebra · 2021
S.-Z. Song et. al. [Linear and Multilinear Algebra. 2016;64(7):1258–1265] studied linear operators from Sm to Sn that preserve cp-rank, where Sn denotes the space of n×n real symmetric matrices. One of the results states that a linear operator ϕ:Sm→Sn preserves cp-rank if and only if ϕ(X)=STXS for some nonnegative matrix S with a full row rank. We show here that while this statement is indeed valid for 2≤m≤3, it is does not hold for m≥4. We conjecture that for n≥4 a linear operator ϕ:Sn→Sn preserves cp-rank if and only if ϕ(X)=STXS for some nonnegative monomial matrix S, and prove the conjecture for 4≤n≤6.