How to invert $(A + \lambda I)$

Question

I would like to simplify the following inverse computation: $$(D+A)^{−1}$$ where $A=U \Sigma U^T$ (eigenvalue decomposition) and $D$ is a diagonal matrix such that $D = \lambda I$

I know the inverse of A is $A^{−1}=UΣ^{−1}U^T$. How could I expand and simplify the inverse calculation ?

Answer 1

Does this help?

$$ (D + A)^{-1} = ( U \Sigma U^T + \lambda I )^{-1} = [ U (\Sigma + \lambda I) U^T ]^{-1} = (U^T)^{-1} (\Sigma + \lambda I)^{-1} U^{-1} = U (\Sigma + \lambda I)^{-1} U^T $$

The inverse of $\Sigma + \lambda I$ can be calculated easily.