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 $X$.

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
  • I want to find non-zero matrix Y. Any suggestion in this direction... – Saheb Dec 15 '17 at 15:49

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.