If S,Z are positive semidefinite. We now know that $y^TSy=\sum_{i}\sum_{j}S_{ij}y_{i}y_{j}\geq 0 $, same goes for Z. We also know that we can write S as $S=\sum_{i}\lambda_{i}x_{i}x_{i}^{T}$, with $\lambda_{i}$ the eigenvalues and $x_{i}$ eigenvectors of the matrix S. In other words $S_{ij}=\sum_{k}\lambda_{k}x_{ik}x_{jk}$ by construction.
I want to show that $\sum_{i}\sum_{j}S_{ij}Z_{ij}\geq0$ Could anyone help me show this??
I would also like to know why the equality of this statement only holds if and only if $S \cdot Z=0$