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 $X$.

  • $\begingroup$ By positive-definite, I assume you mean strictly positive-definite? $\endgroup$ – Yemon Choi Dec 13 '17 at 19:08
  • $\begingroup$ yes it is strictly positive definite matrix $\endgroup$ – Saheb Dec 13 '17 at 20:26
  • $\begingroup$ Related: mathoverflow.net/questions/292946/… $\endgroup$ – 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.

  • $\begingroup$ 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. $\endgroup$ – Saheb Dec 14 '17 at 18:32
  • $\begingroup$ That's another problem.... And it has $Y=0$ as a solution. $\endgroup$ – Federico Poloni Dec 15 '17 at 7:18
  • $\begingroup$ I want to find non-zero matrix Y. Any suggestion in this direction... $\endgroup$ – Saheb Dec 15 '17 at 15:49

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