Let $A\in \mathbb{C}^{ n\times n}$ and $B \in \mathbb{C}^{p \times p}$ be Hermitian matrices with $p < n$. 

Find matrix $X$ such that $X^*AX=B.$

Solution in the case of positive definite $A$ and $B$ was given [here][1].


  [1]: https://mathoverflow.net/a/288454/6818