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!

Why do you think it is always possible? – markvs Jul 20 '20 at 20:30 — markvs, Jul 20 '20 at 20:30
Yes, you are right! I have edited my question. – user3138073 Jul 20 '20 at 20:35 — user3138073, Jul 20 '20 at 20:35

score 2 · Accepted Answer · answered Jul 20 '20 at 20:34

2

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

answered Jul 20 '20 at 20:34

1 Answers1