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.

  • $\begingroup$ Just for completeness: this is usually called a Ricatti equation, sometimes also a Lyapunov equation. This might help to lookup references if needed. $\endgroup$ – leo monsaingeon Jul 20 at 17:10
  • $\begingroup$ Related: mathoverflow.net/q/78106 $\endgroup$ – Rodrigo de Azevedo Jul 20 at 17:28
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    $\begingroup$ Does this answer your question? Solving a quadratic matrix equation $\endgroup$ – vidyarthi Jul 20 at 17:59
  • $\begingroup$ Thank you for the answers, I think Federico's answer is very neat, at least for the type of problem I specified. $\endgroup$ – Fabio Jul 20 at 20:29

$B^{-1/2}XAX^TB^{-1/2}=I$, 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).

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  • $\begingroup$ Thanks a lot Federico! $\endgroup$ – Fabio Jul 20 at 16:00

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