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
I know the inverse of A is $A^{-1}=U\Sigma^{-1}U^T$. How could I expand and simplify the inverse calculation ?
2026-03-29 15:16:48.1774797408
inverse of sum of diagonal matrix and eigendecomposition
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Since $D = \lambda I = \lambda U U^\top$, we have $$ (D+A)^{-1} = (U (\lambda I + \Sigma) U^\top)^{-1} = U (\lambda I + \Sigma)^{-1} U^\top. $$