solve matrix equation

Question

Given symmetric positive definite (SPD) matrices $A$ and $B$, I was wondering there is a solution for

$XAX=B$,

where $X$ is also a SPD matrix.

Thank you!

Answer 1

Your question can be rewritten as $$ (A^{1/2} X A^{1/2})^2 = A^{1/2} B A^{1/2}. $$ So $$ X = A^{-1/2} (A^{1/2} B A^{1/2})^{1/2} A^{-1/2} .$$