Consider a positive-semidefinite Hermitian matrix $A$ of dimension $d\times d$. I would like to know, is it always possible to find a change of basis resulting in a matrix $B$ where the diagonal is a constant? Namely, $B=UAU^\dagger$ with $U$ invertible has $B_{ii}=c$.
If in general such a $U$ does not exist, for which $A$ does it?
P.S. I hope my question is well posed, otherwise I can make edits to state the problem more precisely and concretely.
EDIT 1: what if $U^\dagger U = 1$ ?
The answer is yes. In particular, see corollary 5 of this document.