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
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 .