TITLE: How can I prove $A − A(A + B)^{−1}A = B − B(A + B)^{−1}B$ for matrices $A$ and $B$?
QUESTION [7 upvotes]: The matrix cookbook (page 16) offers this amazing result:
$$A − A(A + B)^{−1}A = B − B(A + B)^{−1}B$$
This seems to be too unbelievable to be true and I can't seem to prove it. Can anyone verify this equation/offer proof?

REPLY [3 votes]: You may simply put $X=A+B$ and show that
\begin{aligned}
A-AX^{-1}A
&=(X-B)-(X-B)X^{-1}(X-B)\\
&=(X-B)-(X-2B+BX^{-1}B)\\
&=B-BX^{-1}B.
\end{aligned}