Trigonalizable endomorphism/Jordan normal form/Fact/Proof

Proof

Since is trigonalizable, we can apply fact. Hence, there exists a direct sum decomposition

where the generalized eigenspaces are -invariant. Looking at the situation for each generalized eigenspace, we may assume that has only one eigenvalue , and that

holds. Then,

is nilpotent. Therefore, because of fact, there exists a basis such that is described by a matrix of the form

where the equal or equal . With respect to this basis,

has the form