Assume that $X$ is a full rank $n \times k$ matrix and that $A$ is $n\times n$ non-singular matrix positive semidefinite. I would like to show that $A^{-1} - X(X'AX)^{-1}X'$ is positive semidefinite.
I tried to use the singular decomposition of $X$ as suggested in this similar post. But I can't get the result.
I suppose that $X'$ means $X^\ast$. Your statement is false in general. Since $f(A)=A^{-1}-X(X'AX)^{-1}X'$ is not identically zero, if $f(A)$ is nonzero and positive semidefinite, then $f(-A)$ is nonzero and negative semidefinite.
The statement is true when $A$ is positive definite, however. Let $A=LL'$ (e.g. take $L$ as $A^{1/2}$ or the Cholesky factor of $A$) and let $Y=L'X$. Then $$ A^{-1}-X(X'AX)^{-1}X'=(L^{-1})'\big(I-Y(Y'Y)^{-1}Y'\big)L^{-1}. $$ Since $L$ is nonsingular and $I-Y(Y'Y)^{-1}Y'$ is an orthogonal projection, the LHS of the above is positive semidefinite.