# A conjugation matrix $X\in \mathbb{C}^{ n\times p}$ where $p< n$

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 the matrix $X$.

• By positive-definite, I assume you mean strictly positive-definite? – Yemon Choi Dec 13 '17 at 19:08
• yes it is strictly positive definite matrix – Saheb Dec 13 '17 at 20:26
• – Federico Poloni Feb 14 '18 at 13:20

Let $A=R^*R, B=S^*S$ be Cholesky factorizations. Then for any orthogonal $Q\in\mathbb{C}^{n\times n}$ the matrix $X=R^{-1}Q\begin{bmatrix}S\\0\end{bmatrix}$ works, as can be verified directly.
• Basically I want to find a matrix $Y\in \mathbb{C}^{n \times p} (p <n)$ such that the matrix $Y^HAYP$ is a Hermitian matrix where $A\in \mathbb{C}^{ n\times n}$ is a Hermitian positive definite matrix and $P\in \mathbb{C}^{ p\times p}$ is an arbitrary matrix. Please help me in solving this problem. – Saheb Dec 14 '17 at 18:32
• That's another problem.... And it has $Y=0$ as a solution. – Federico Poloni Dec 15 '17 at 7:18