Range of Weak Values: Explaining Eigenvalues of A

[msumm21]

The weak value of an observable A with pre-selected state $\psi$ and post-selected state $\phi$ is defined as:
A_w =\langle \phi | A | \psi \rangle / \langle \phi | \psi \rangle

References like Wikipedia then include a comment that the weak value is not bounded by the range of eigenvalues of A, e.g. when $\phi$ and $\psi$ are nearly orthogonal the value gets larger than the biggest eigenvalue of A. I know the denominator is small in such cases, but when I do the math it seems like the numerator should shrink in proportion and the range of the weak values should not exceed the range of the eigenvalues of A, let me explain and please let me know if you see my mistake.

If I rewrite $\psi$ as a linear combinations of the eigenstates of A, e.g.
\psi = \sum_i \psi_i |A_i \rangle
then do the math I just end up with a weighted average of the eigenvalues (\alpha_i) of A, i.e.
A_w = \sum_i w_i \alpha_i / \sum_i w_i where
w_i = \psi_i \langle \phi | A_i \rangle

Anyone see what I missed?

 

[Demystifier]

You miss the possibility that absolute values of w_i are not small, but have different signs (or phases if w_i are complex) such that \sum_i w_i is small.

 

[msumm21]

Ah yes. Thanks Demystifier!