Find the matrix $X\in \mathbb{C}^{ n \times p}$ where $p<n$

Question

Given a Hermitian positive definite matrix $A\in \mathbb{C}^{n\times n}$ and a Hermitian matrix $B\in \mathbb{C}^{p \times p}$, find the matrix $X$ so that $X^HAX=B$ holds where $X^H$ denotes conjugate transpose of $X.$

Answer 1

You need $B$ to be positive semidefinite as well. Then let $X = A^{-1/2} B^{1/2}$, where $A^{-1/2}$ and $B^{1/2}$ are the positive (semi) definite square roots of $A^{-1}$ and $B$.