If I have two positive definite Hermitian NxN matrices A and B, if I adiabatically change the components of A to B (constraining any intermediate matrices to be Hermitian as well, but not necessarily positive definite) while \"following\" the eigenvalues ... will the mapping of the eigenvalues of A to the eigenvalues of B be independent of the adiabatic path? Obviously the answer is yes for N=1. And I have convinced myself the answer is yes for N=2 (given that the path doesn\'t take you through the zero matrix). But I am not sure how to approach this in a general enough manner to figure this out. Can anyone here offer some advice? I eventually want to find the mapping of the eigenvalues of A to B. So if this is a solved problem, by all means point me to the solution so that I may read up on it.