G and V are two positive definite symmetric matrices. How to find a symmetric matrix W such that:

$$W G W =V$$

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    $\begingroup$ who says this is possible? $\endgroup$ – Will Jagy Sep 19 '15 at 16:30
  • $\begingroup$ I read a paper "Quantum source of entropy for black holes " by Luca Bombelli, Rabinder K.Koul , Joohan Lee, and Refael D. Sorkin. The last equation in second page is what I asked. In my calculation, It is only possible when all the eigenvalues of G are the same. $\endgroup$ – xjtein Sep 20 '15 at 2:13
  • $\begingroup$ I don't have the article. You had better typeset the last equation from the second page in your original question, and describe as carefully as possible the conditions they are using. Hypotheses are everything. $\endgroup$ – Will Jagy Sep 20 '15 at 2:16
  • $\begingroup$ Oh, if you do not think it is always possible, you should stress that. $\endgroup$ – Will Jagy Sep 20 '15 at 2:19
  • $\begingroup$ I send you a copy of the paper. Wish you can help me. $\endgroup$ – xjtein Sep 20 '15 at 2:25

That works. given a real symmetric and positive definite matrix, we can find a real symmetric square root: given $G,$ find orthogonal $P$ such that $P^T G P = D$ is diagonal and positive. So $P D P^T = G.$ Let $\sqrt D$ be the diagonal matrix with entries the (positive) square roots of the relevant entries of $D.$ Then $\left( P \sqrt D P^T \right)^2 = P D P^T =G.$

So we can find symmetric positive definite $H$ with $$ H^2 = G. $$

Next, find symmetric positive $U$ such that $$ U^2 = H V H. $$

Let $$\color{red}{ W = H^{-1} U H^{-1}}. $$

Confirm $$ W G W = H^{-1} U H^{-1} H^2 H^{-1} U H^{-1} = H^{-1} U U H^{-1} = H^{-1} H V H H^{-1}= V $$


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