Monotocity of a trace function

Question

Define a function on the space of n-dimensional positive definite matrices: $$ f(X)=tr(X^{-1}-(X+M)^{-1})， X>0, M>0\text{ a constant matrix}. $$ prove that $f$ is monotonous, i.e. $$ X\geq Y>0 \Rightarrow f(X)\leq f(Y) $$

Answer 1

The claim does not hold. For it to hold, we should have $\nabla f(X) \le 0$ (in Löwner ordering). That is, \begin{equation*} (X+M)^{-2}-X^{-2} \ge 0, \end{equation*} should hold for all $X>0$.

Take for example the strictly positive definite matrices:

\begin{equation*} X = \begin{bmatrix}17 & -18\\ -18 & 45 \end{bmatrix},\qquad M =\begin{bmatrix}29 & -17\\-17 & 10\end{bmatrix}. \end{equation*} Then, $(X+M)^{-2}-X^{-2}$ is a matrix will all entries negative (elementwise), hence it not positive definite.