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All matrices are $n \times n$. $C$ is real symmetric positive definite.

How to solve $C = X^\top C X$ for $X$?

I am interested in characterizing both the set of real matrices satisfying the equation and the (possibly larger) set of complex solutions.

ziutek
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1 Answers1

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According to Choleski, $C=B^TB=X^TB^TBX=Y^TY$ with $Y=BX$. Then it remains to solve $B^TB=Y^TY$. Cf. my posts, for the real case, in General Cholesky-like decomposition and, for the complex case, in How to get matrix $A$ from $A^\top A=B$ with given symmetric matrix $b$?

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