I'm currently taking quantum mechanics and diagonal matrices, along with the idea of diagonalization, comes up alot. One line in my textbook threw me off completely and made me realize I don't understand these matrices at all.
This is in regards to the operating on a Hydrogen state with the relativistic perturbation kinetic energy Hamiltonian:
"Thus, although the eigenstates of the unperturbed Hamiltonian are highly degenerate, the matrix representation of the perturbing Hamiltonian in each degenerate subspace is already diagonal, and we can calculate the first order energy shift by simply taking the expectation value."
This is just for context. What I'm hoping to get here is a nice explanation of what diagonalization means and if possible, some good resources where I can read further.