Inverse of the sum of a invertible matrix with known Cholesky-decomposion and diagonal matrix

Question

I want to ask a question about invertible matrix. Suppose there is a $n\times n$ symmetric and invertible matrix $M$, and we know its Cholesky decomposion as $M=LL'$. Then do we have an efficient way to calculate $(M+D)^{-1}$, where $D=diag(d_1,...,d_n)$ with positive diagonal entries, by taking the information of $M=LL'$ rather than calculating from scratch with $M+D$ directly? Or what if for the sepcial case $D=dI_n$?

Thanks a lot!

Answer 1

I'm not sure how to use Cholesky here but here is a method that avoids inverting non-diagonal matrices.

1. First write (skip this step for $D=dI$), $$(M+D)^{-1}=D^{-1}(MD^{-1}+I)^{-1}$$
2. Now use the fact that real symmetric matrices are (orthogonally) diagonalizable, and so you can find $MD^{-1}=Q\Lambda Q^T$.
3. Then,

$$=D^{-1}(Q\Lambda Q^T+I)^{-1}=D^{-1}(Q(\Lambda+I)Q^T)^{-1}=D^{-1}Q(\Lambda+I)^{-1}Q^T.$$