A nonzero vector can't correspond to two different eigenvalues of A.
True, can only correspond if the eigenvalue is of multiplicity > 1.
Correct?
What do you mean by saying that a "non-zero vector x corresponds to an eigenvalue of A"? Du you mean that holds (i.e. that x is an eigenvector of A for the eigenvalue )?
And is what you wan't to show that, from x being non-zero and that and , it follows that ?
If so, this has got nothing to do with multiplicities of those eigenvalues:
.