I'm reading Nielsen and Chuang, and I'm trying to understand what we actually mean when we talk about observables and their relationship to measurements. I'm more of a mathematician than a physicist, and I think there are intuition disconnects when it comes to the intuition/meaning behind these concepts.
When I think about a state $|\psi \rangle = a|0 \rangle + b|1 \rangle$ that gets measured, I imagine that we have a lab and some equipment that measures some property of a quantum object, which causes the state to collapse to $|0 \rangle$ with probability $a^2$. In particular, we can explicitly measure/observe/infer that the object is in state $|0 \rangle$ or $|1 \rangle$ when we use our measurement equipment.
Now, when we talk about a measurement observable we have some Hermitian matrix $M=\sum m P_m$, where $P_m$ are matrices associated with the spectral decomposition of $M$ and $m$ are the eigenvalues. When we "measure" with respect to this, what does this exactly mean? I understand that each eigenvalue is associated with the resulting state $|\psi' \rangle \propto P_m |\psi \rangle$, but what does this mean physically? When we measure our quantum object's state with our lab equipment, we're not measuring the eigenvalues of a matrix, are we? What does it mean to get back an eigenvalue as the result of a measurement, in lab-and-equipment language?