I came across the following relationships, but I have no idea how to prove them. I would love to know they can be proved.
Suppose $X$ and $Y$ are both symmetric matrices,
relationship: $$(X + Y)^{-1} = X^{-1} - X^{-1}(X^{-1} + Y^{-1})^{-1} X^{-1}$$ $$(X + Y)^{-1} = Y^{-1} - Y^{-1} (X^{-1} + Y^{-1})^{-1}Y^{-1}$$ $$(X + Y)^{-1} = X^{-1}(X^{-1}+Y^{-1})^{-1}Y^{-1}$$
I would like to know these relationships can be proved. Thanks
We use following identity to prove the claims: $$ A^{-1}B^{-1}=(BA)^{-1}. $$ Start with the last one: $$ X^{-1}(X^{-1}+Y^{-1})^{-1}Y^{-1}=((X^{-1}+Y^{-1})X)^{-1}Y^{-1}\\ =(I+Y^{-1}X)^{-1}Y^{-1}\\ =(Y(I+Y^{-1}X))^{-1}\\ =(Y+X)^{-1}. $$ We use this to prove the first identity: $$ (X + Y)^{-1} +X^{-1}(X^{-1} + Y^{-1})^{-1} X^{-1}= (X + Y)^{-1} +(X+Y)^{-1} Y X^{-1}\\ = (X + Y)^{-1}(I+Y X^{-1})\\ = (X + Y)^{-1}(X+Y) X^{-1}= X^{-1}\\ $$ The second identity follows by the same argument.
Remark: I did not see how symmetry assumption can be used. The only thing that is needed is $A,B$ being of same dimension and having inverses.