I have a symmetric positive definite 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 Block LU decomposition. However, I couldn't figure out how to generalize this to n dimension.
Thanks for your help.
Bin