# LU Decomposition of PSD Matrix + Diagonal Matrix

If I have a psd, symmetric matrix $$\mathbf{A}$$ and I need to do LU decomps on $$\mathbf{B_i}= \mathbf{A} + \mathbf{D_i}$$ (where $$\mathbf{D_i}$$ is a diagonal psd matrix, where $$\mathbf{D_i}$$ changes randomly). Is there an easy way to go from the LU of $$\mathbf{A}$$ + some function of $$\mathbf{D_i}$$ to get the LU of $$\mathbf{B_i}$$?

Any references in the literature would be greatly appreciated.

## 1 Answer

Use a Cholesky decomposition or an LDL decomposition instead of LU.

Judging from answers to:

there's no good way to update any of those decompositions. You'd do no worse to recompute the decomposition from scratch each time unless the rank of the diagonal matrix is really low, in which case, maybe you could do something similar to Sherman-Morrison(-Woodbury)-type updates.