Inverse of matrix plus identity

Question

I have a matrix $A$ which is symmetric and positive definite, and I am curious about the properties of $(I+A)^{-1}$.

I can tell that the matrix will exist (that is, $I+A$ will be symmetric and invertible), and thus that $(I+A)^{-1}$ will also be symmetric. I am curious if there are any other known properties of this inverse.

For example, is there anything we can say about the products $A(I+A)^{-1}$ or $(I+A)^{-1}A$?

Answer 1

For example, one can say

$A(I + A)^{-1} = (A^{-1})^{-1}(I + A)^{-1}$ $= ((I + A)A^{-1})^{-1} = (A^{-1} + AA^{-1})^{-1} = (I + A^{-1})^{-1}. \tag 1$

It is also easy to see that

$(I + A)^{-1}A = (I + A^{-1})^{-1}. \tag 2$

Answer 2

$I+A$ has the same eigenvectors as $A$. So the eigenvalues of $(I+A)^{-1}$ are $(1+\lambda_i)^{-1}$, where the $\lambda_i$ are the eigenvalues of $A$.