Here is some quick note for the existence of eigenvector for linear operator in complex vector space of dimension . Consider any non-zero vector in the space, we have the vectors to be linearly dependent. Thus, there exists such that . If is zero, we will pick the largest , , such that . In any case, we can write

with .

Now consider the polynomial expression

. We can find complex numbers such that

. Easy to verify that

.

For the above to be true, there must exist an , , such that

but . Define , note that with . That is and thus and are precisely an eigenvalue and eigenvector of .