Based on the [Woodbury identify][1] we know that:
$(A + B)^{-1}$ = $A^{-1} - A^{-1}(AB^{-1} + I)^{-1}$
It can also be shown that if $B$ is PSD,
$(A + B)^{-1} \leqslant A^{-1}$
Therefore if $B = cI$,
$(A + cI)^{-1} \leqslant A^{-1}$
This is a fairly weak bound, so Im wondering if we can somehow create a tighter bound that uses information in $B$ such as $c$ when $B = cI$. For example when $B-cI$, the Woodbury identify can be re-arrange as follows, but not sure if this helps..
$(A + B)^{-1}$ = $A^{-1}(cA^{-1} + I)^{-1}$
Any help to tighten these two bounds, both when $B$ is any PSD matrix, and when $B=cI$ would be appreciated.