In Susskind's Quantum Mechanics book The Theoretical Minimum, Eqn 3.21 on page 82 states that $\sigma\_z |r\rangle=\frac{1}{\sqrt2}|u\rangle-\frac{1}{\sqrt2}|d\rangle$. He is clearing up confusion as to whether the operator acting on a state results in the state that we find upon measurement. He says this is definitely not the case since the $\sigma\_z$ measurement would result in either +1 or -1 (so the state would either be $|u\rangle$ or $|d\rangle$, respectively). This makes sense.

Instead, it is suggested that the interpretation is that the $\frac{1}{\sqrt2}|u\rangle-\frac{1}{\sqrt2}|d\rangle$ is used simply for the calculation of probabilities (1/2 for $|u\rangle$ and 1/2 for $|d\rangle$ by squaring the magnitude of each corresponding coefficient). This still makes sense.

This makes me think of the eigenvalue equations: $\sigma\_z|u\rangle=|u\rangle$ and $\sigma\_z|d\rangle=-|d\rangle$. The above paragraph is consistent with this as well since the probabilities will simply be 1 for discovering it to be in the state it is already in.

My question is this: For a general eigenvalue equation $H|\psi\rangle=E|\psi\rangle$, what happens to this interpretation if, say, $E=2$ (or anything greater than 1)? If we measure the observable $H$ while the system is in state $|\psi\rangle$, I suppose the interpretation suggests there is a 400% probability ($2\cdot2$) of discovering that the system is in state $|\psi\rangle$.

This interpretation seems to be very special to the Pauli matrices since their eigenvalues are only $\pm1$. I assume there is a simple explanation (do we have to normalize something?) but the book seems not to address it. Or maybe I am misunderstanding the interpretation altogether.