UCSD's online QM notes, as usual, starts stating that QM operators are Hermitian and says that operator $O$ elements can be computed by
$$O_{ij} = \langle u_j|O|u_i\rangle$$
The $u_i$ are eigenvectors, which are orthogonal to each other, thanks to being Hermitian. So, I suppose that $O$ is a Hermitian operator. But where are the eigenvalues?
I expect that when you apply the operator to its eigenvector, a $\lambda$ must appear. Yet, I see it nowhere. Neither, text says that $O$ is a diagonal, as I may expect if I apply $\langle u_i|$ to $O|u_i\rangle = \lambda_i |u_i\rangle$, I should get $\lambda_i$ on the main diagonal and 0 everyelse. Are $u_i$ eigenvectors of the $O$ or some another operator?