What is the significance of diagonal matrices in Quantum Mechanics?

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.

Diagonal matrices are useful in many ways and it would take long to compile a list of all of them. I would like to point out why it is useful in your case.

When dealing with the Schrödinger equation the eigenstates of the time-independent Hamiltonian are important since they are the stationary states of the time-dependent equation this means that over time they only acquire a phase factor. I.e. if $$\hat{H}\phi = E \phi$$ then:

$$\phi(t) = e^{iEt} \phi(0)$$

You can solve the first of these equations easily when the Hamiltonian is diagonal (try it).

Another way to see this is to note that the more general form of the second equation, which applies to any state can be written in operator form:

$$\phi(t) = e^{it\hat{H}} \phi(0)$$

where the exponential is defined through its power series. Now if H is diagonal all powers of H are just the entries squared which simplifies the formalism.

A diagonal matrix is one that is expressed in the basis of its eigenstates. Remember, a matrix has many equivalent representations depending on the basis you use. Thus, whenever your matrix is diagonal then its eigenstates can be represented as a column vector with one 1, and the rest of the entries zero. Say the column vector you chose has a 1 in the $k$-th row, then act on this vector with your diagonal matrix, what will be the product? It will be the same vector (hence it is an eigenstate) multiplied by the number in the $k$-th column and $k$-th row of your diagonal matrix (which is the eigenvalue of that eigenvector).

But eigenvectors represent states of a system after measurement! Thus, measuring a system's, say, energy, will cause it to be in an eigenstate of the energy operator, with the corresponding eigenvalue as its energy. That's why diagonal matrices are important in Quantum Mechanics: they give us a straightforward view of what state our system can be in upon measurement, with the corresponding energy of that state, all in a nice diagonal matrix to read from!