Dear all, 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
Cholesky decomposition and permutation matrix
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matrices
1 Answers
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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).
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0@Fabian: actually I spent several days trying to find the answer by myself, in vain, before I came here to ask. – 2011-05-05