Let $A$ be a singular matrix. I would like to understand the behavior of $A^{+}$ compared to $(A+\epsilon I)^{-1}$.
The reason I'm asking is because I want to understand $MB^{+}M^{*}$ if $M=(AB)^{1/2}$.
If $B$ were nonsingular, then $$ MB^{-1}M^{*} = (AB)^{1/2}B^{-1}(BA)^{1/2}=B^{-1}B(AB)^{1/2}B^{-1}(BA)^{1/2} = B^{-1}(BA)^{1/2}(BA)^{1/2}=B^{-1}BA=A. $$
I'd like to show, for the same choice of $M$, that $MB^{+}M^{*} \leq A$ when $B$ is singular. This is clearly the case if it's nonsingular, but not sure how to proceed otherwise.
Now I have $$ MB^{+}M^{*} = (AB)^{1/2}B^{+}(BA)^{1/2} $$ and is there any way to make sense of this compared to $M(B+\epsilon I)^{-1}M^{*}$ using a continuity argument?