Linear Algebra Done Right · Problem 8B.9

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · generalized-eigenspace-decomposition-jordan

Problem

Suppose

\mathbf{F} = \mathbf{C}

and

T \in \mathcal{L}(V)

. Prove that there exist

D, N \in \mathcal{L}(V)

such that

T = D + N

, the operator

D

is diagonalizable,

N

is nilpotent, and

DN = ND