For context I was taking an optimization course and at one point we used the restriction by line to prove the concavity of $\log det(X)$. Let $g(t) = \log det(X + tV)$ with $V \in S_{n}$ and $t \in \mathbb{R}$.
This leaves us with $X + tV \in S_{n}$ then decompose it as follows: $$X + tV = X(I + tX^{-1}V) = X(I + tX^{-\frac{1}{2}}X^{-\frac{1}{2}}V) = X(I + tX^{-\frac{1}{2}}VX^{-\frac{1}{2}})$$
And here's the problem: I can't see the justification (which must be clear, since it's not spelled out) for being able to commute $X^{-\frac{1}{2}}$ with $V$. I know that if the product of symmetrical matrices is symmetrical then they commute, but here we don't know if the matrix $X^{-\frac{1}{2}}V$ is symmetrical.
I often see commutations that I do not understand and that are not justified, so maybe there are rules I don't know (especially when working with symmetric or positive definite matrices).
Thanks a lot!