Let's assume that I have a symmetric matrix $\Sigma$ and a permutation matrix $A$. Is there a relationship between the Cholesky decompositions of $\Sigma$ and of $A^T \Sigma A$? Many thanks, Samuel.
[Math] Cholesky decomposition and permutation matrix
cholesky decompositionmatricespermutation-matrices
Best Answer
The Cholesky decomposition allows you to write (for real symmetric positive definite matrix $\Sigma$) $$\Sigma = U^T U$$ with $U$ an upper triangular matrix. Therefore, $$A^T \Sigma A = A^T U^T U A = B^T B$$ with $B= UA$. Note however that $B$ is (in general) not upper triangular anymore such that $B^T B$ is not the standard Cholesky decomposition (but of course very similar to a Cholesky decomposition).