I'm wondering if a transcendental function of a PSD matrix can have different eigenvectors than the original matrix. For a polynomial $f()$, it is clear that the $f(S)$ for a PSD matrix $S$ would have the same spectrum and possibly different eigenvalues. This can be seen by writing the spectral decomposition and doing simple manipulation.
I am not sure if this works for transcendental functions. Intuitively, we can write the function in an infinite series where it converges. Since it converges, I can look at the sum of the first $K$ terms of the expansion and see that the eigenvectors are the same - and carry over this argument for all $K$. However, I am not sure how rigorous the above argument is and if the claim is actually true.