Why are the only possible values of an operator its eigenvalues?

[randomafk]

Like the title says, why are the only possible values of an operator its eigenvalues?

reading shankar right now and I'm having difficulty understanding why this has to be the case, given some operator/variable Ω

 

[bossman27]

This conversation on a stackexchange thread looks pretty good and there are multiple answers which go into some depth: http://physics.stackexchange.com/questions/8156/eigenvalues-of-an-operator-correspond-to-energy-states-in-quantum-mechanics-why

 

[atyy]

It is a postulate that happens to work.
http://homes.cs.washington.edu/~oskin/quantum-notes.pdf
http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html