...but are they forced to have one that is 0? Alternatively, are non-bijective linear maps forced to have at least one eigenvalue?
I'm asking as apparently for a finite dimensional vector space, such that is nilpotent (there exists such that ) then zero is the only eigenvalue. It is clear that zero is the only possible eigenvalue, but I'm unsure if it is indeed an eigenvalue. (Note that if is nilpotent that it is not a bijection...which I hoped would be the route to the solution...)