I made a small but significant mistake in my earlier post. The vector
is supposed to be non-zero.
The existence of an eigenvalue implies the existence of an eigenvector. With the definitions from my earlier post it's immediate. What does it mean that
has an eigenvalue? This:
1)
What does it mean that
has an eigenvector? This:
2)
These formulas are equivalent. The order of the existential quantifiers doesn't matter.
However, Sheldon Axler uses a different definition of an eigenvalue. For him, that
has an eigenvalue means this:
3) There exists
such that
is not injective.
We want to see that 3) implies 2). What does it mean that
is not injective? It means that we have two vectors
such that
and
the last formula being equvalent to
and further equivalent to
Since
we have that the vector
is non-zero. Therefore, we have found a non-zero vector
such that
just as we wanted.