# Given the Cholesky decomposition of $A$, compute efficiently Cholesky decomposition of $RAR^T$?

Let $$B = RAR^T$$, where $$A$$ is positive definite and symmetric, and $$R$$ a generic matrix (possibly rectangular).

Suppose I know the Cholesky decomposition $$A=LL^T$$. Is it possible to compute the Cholesky decomposition of $$B$$ fast, exploiting the knowledge of $$L$$?

Edit: Let's assume that we also know the an LU decomposition for $$R=L_RU_R$$. I'm not sure if this can help to obtain the decomposition of $$B$$.

• For one, note that $B=RLL^TR^T=(RL)(RL)^T$. So, you just need to make $RL$ lower triangular. – cmk May 18 at 18:54
• @cmk But $RL$ is not lower triangular in general. Therefore this is not a Cholesky decomposition of $B$. Some more steps are required. – becko May 18 at 18:55
• Right, but now all you need to do is make the product $RL$ lower triangular, and you'll have the desired decomposition. – cmk May 18 at 18:56
• @cmk Ok, but how? – becko May 18 at 18:57
• There are various algorithms to reduce a matrix to lower triangular form. The thing is most of them are computationally comparable to the Cholesky factorization, so I'm not sure if this process is really that much more efficient; it just seems like the obvious thing to do for the problem. – cmk May 18 at 19:10