Problem Statement: Suppose is such that every is an eigenvector of . Prove that is a scalar multiple of the identity operator.
Proof: Let . Then by assumption there exist scalars such that and . Now consider . Since is a vector space it follows that and so there exists a scalar, , such that . But is a linear operator and so it follows that and so setting these two equations equal we see that . Equating like terms we get and and so it follows that . Thus, for any , for some scalar .
Now we must consider if are already scalar multiples of each other. Consider: and . Then . An argument similar to what was done above to show that will show that all of these scalars are in fact the same scalar.
Thus, is a scalar multiple of the identity operator.