Can someone please explain to me how for $Z$, $V \epsilon S^n$, $t \epsilon R$ $(Z+tV) = Z^{1/2}(I + tZ^{-1/2}VZ^{-1/2})Z^{1/2}$
I think I'm missing some fundamental aspect of matrix algebra. I've been using the matrix cookbook for help
edit: I removed the first part because in typing it out for the question, I realized my mistake!