I am looking for conditions to have a matrix $Z=X\oplus Y$, where $\oplus$ denotes the Hadamard product (with an abuse of notation), to have $rank(Z)=1$. Note that X and Y are both positive semi-definite matrices.
I know for instance that $rank(Z)\leq rank(X)rank(Y)$ and any real positive semi-definite rank-1 matrix $Z$ of size $n$ can be written as $Z=xx^T$, where $x$ is some vector in $\mathbb{R}^n$. However, they do not really help.