S.-Z. Song et. al. [Linear and Multilinear Algebra. 2016;64(7):1258–1265] studied linear operators from ${\mathcal{S}}_m$ to ${\mathcal{S}}_n$ that preserve cp-rank, where ${\mathcal{S}}_n$ denotes the space of n × n real symmetric matrices. One of the results states that a linear operator $\varphi :{\mathcal{S}}_m\to {\mathcal{S}}_n$ preserves cp-rank if and only if ϕ(X) = S^TXS 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 $\varphi :{\mathcal{S}}_n\to {\mathcal{S}}_n$ preserves cp-rank if and only if ϕ(X) = S^TXS for some nonnegative monomial matrix S, and prove the conjecture for 4 ≤ n ≤ 6.

S.-Z. 宋等。阿尔。[线性和多重线性代数。2016;64(7):1258–1265] 研究了来自${\mathcal{小号}}_{米}$到${\mathcal{小号}}_n$保留 cp-rank，其中${\mathcal{小号}}_n$表示n  ×  n实对称矩阵的空间。其中一个结果表明线性算子$\phi :{\mathcal{小号}}_{米}\to {\mathcal{小号}}_n$保留 cp-rank 当且仅当ϕ ( X ) =  S ^T XS对于某些具有完整行秩的非负矩阵S。我们在这里表明，虽然这个陈述对于 2 ≤ m ≤ 3确实有效  ，但它不适用于m  ≥ 4。我们推测对于n  ≥ 4 一个线性算子$\phi :{\mathcal{小号}}_n\to {\mathcal{小号}}_n$保留 cp-rank 当且仅当ϕ ( X ) =  S ^T XS对于某个非负单项式矩阵S ，并证明 4 ≤ n  ≤ 6的猜想 。