Let $\mathbf{A}$ a $n\times n$ symmetric positive definite matrix and $\alpha$ a positive constant. I want to simplify the following expression: $$\left(\mathbf{A} + \alpha \, \mathbf{I}\right)^{-1},$$ where $\mathbf{I}$ is the identity matrix of order $n$.
I looked in the Matrix Cookbook to find an identity, but I found only more general formulas like the Woodbury identity. Do you know how to find a simpler identity for this easy case?
Thanks a lot!
Edit: my goal is to obtain an easy expression in terms of products of $\mathbf{I}$, $\mathbf{A}$, $\mathbf{A}^{-1}$, $\alpha$ and $\alpha^{-1}$ without computing other inversions or decompositions. I can use the same term more times and I don't have to use all of them (I do not want to compute redundant operations).