Sylvester's law of inertia states given a symmetric matrix $A$ and a squared invertible matrix $S$ of the same size, then $A$ and $SAS^\top$ have the same number of positive, negative, and zero eigenvalues.
Question: I'm wondering if this theorem can be generalized to the case when $S$ is not a squared matrix. For example, $S\in\mathbb{R}^{n\times m}$ with $n<m$, and $rank(S)=n$.