Let $d \times d$ matrices $A, B$ be positive definite. Is there a closed form solution for the following quadratic equation in $X$?


$$X A X^{T} = B$$


Thank you.



Closed form solution for $XAX^{T}=B$
$B^{-1/2}XAX^TB^{-1/2}=I$, so $B^{-1/2}XA^{1/2}=Q$ must be orthogonal. On the other hand, for any orthogonal $Q$, it is simple to verify that $X = B^{1/2}QA^{-1/2}$ solves the equation, so this is a complete parametrization of the solutions.


Here $A^{1/2}$ is the symmetric square root of $A$ (if you prefer you can work with the Cholesky factor and obtain similar results).


