As far as I gather, before a perturbation is applied, the eigenspace associated with the degenerate energy is multidimensional but after applying the perturbation this space 'splits' into different eigenspaces. If we use Non-degenerate perturbation theory, then we would end up dividing by zero when calculating the coefficients for the linear combination of the perturbed state. The way to get around this is apparently to diagonalise the perturbation Hamiltonian with a basis of the unperturbed degenerate eigenstates, and this is what I fail to understand.
Why is it that diagonalising the perturbation Hamiltonian will solve the problem of dividing by zero? Also how can I understand visually what is happening to the linear transformation on the eigenspace before and after diagonalising the perturbation Hamiltonian?
Another problem I have is understanding the process of calculating the energy and state corrections after finding a basis in which the perturbation Hamiltonian is diagonalised. Every example or explanation I have found is incredibly abstract. I would love to see a simple example with matrices and vectors if possible if anybody has links to good resources or can be bothered to go through one in an answer. I find matrix/vector representations to be much more intuitive.
Lastly, another question that I have just thought of: Are these eigenstates that we find to diagonalise the perturbation Hamiltonian with precisely the eigenstates of the new total Hamiltonian? And in that case, are they regarded as the first order corrections to the space of eigenstates whatever that would mean?