Linear Algebra Done Right · Problem 8B.5

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · generalized-eigenspace-decomposition

Problem

Suppose

T \in \mathcal{L}(V)

and

3

and

8

are eigenvalues of

T

. Let

n = \dim V

. Prove that

V = (\operatorname{null} T^{n-2}) \oplus (\operatorname{range} T^{n-2})