Diagonal and block diagonal matrix What's the significance of diagonal and block-diagonal matrices in quantum mechanics? For instance, let $S$ be the symmetry operator, since $[S,H]=0$, they have a shared eigenbasis. If I use a basis in which S is diagonal, H will be block diagonal. What does this mean? Thanks!!
 A: When we diagonalise a matrix we are in a process of determining it's Eigenvalues (and Eignevectors). It is similar story for block-diagonal matrices. The difference between diagonal and block-diagonal matrix forms
lies in the notion of degenerate Eigenstates. That is, different states with similar Eigenvalues (in this case, energies). If we have an operator $S$ which commutes with the Hamiltonian $H$ i.e, $ [S,H] = 0$ we can diagonalize them first to find a simultaneous set of Eigenstates for these operators (and $S$ does  indeed represent a symmetry operation which leaves $H$ invariant) and the matrix represented by the similarity transformation $S^{-1}H S $ is diagonizable.
In the case where we find a set of Eigenstates that are in fact degenerate states, we then look for Eigenstates of the Hamiltonian that are a linear combination of the states in that (degenerate) space of states. In this new basis of states, the matrix of the Hamiltonian is block diagonal (a square diagonal matrix in which the diagonal elements are also square matrices).
