# Cholesky decomposition for sparse matrix

I have a matrix that is composed of small block diagonal matrices. For example:

$$M = \left[ \begin{array}{ccc} \Sigma & \Psi & \Psi \\ \Psi & \Sigma & \Psi \\ \Psi & \Psi & \Sigma \\ \end{array} \right],$$ where both $\Sigma$ and $\Psi$ are diagonal matrices.

My question is: Is there an efficient way to perform Cholesky decomposition on this? It is such a regular matrix, that I feel like one must be able to do some trick to simplify the problem, rather than use brute force method.

For 2x2, I found a simple solution shown here (http://en.wikipedia.org/wiki/Block_LU_decomposition). However, I couldn't figure out how to generalize this to n dimension.

Thanks for your help.

Bin

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I presume it is symmetric positive definite? –  copper.hat Jun 16 '12 at 1:05
Yes, that's correct. Sorry I forgot to mention that. –  Bin Jun 17 '12 at 17:41
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