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 $X$.
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$\begingroup$ By positive-definite, I assume you mean strictly positive-definite? $\endgroup$– Yemon ChoiCommented Dec 13, 2017 at 19:08
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$\begingroup$ yes it is strictly positive definite matrix $\endgroup$– SahebCommented Dec 13, 2017 at 20:26
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$\begingroup$ Related: mathoverflow.net/questions/292946/… $\endgroup$– Federico PoloniCommented Feb 14, 2018 at 13:20
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1 Answer
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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.
answered Dec 13, 2017 at 21:52
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$\begingroup$ 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. $\endgroup$– SahebCommented Dec 14, 2017 at 18:32
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$\begingroup$ That's another problem.... And it has $Y=0$ as a solution. $\endgroup$ Commented Dec 15, 2017 at 7:18
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$\begingroup$ I want to find non-zero matrix Y. Any suggestion in this direction... $\endgroup$– SahebCommented Dec 15, 2017 at 15:49