A question about positive definite matrix decomposition

Question

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

$$W G W =V$$

Answer 1

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