Linear Algebra Done Right · Problem 8A.15

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · nilpotent-operators-diagonalizable

Problem

Suppose

\mathbf{F} = \mathbf{C}

and

T \in \mathcal{L}(V)

. Prove that

T

is diagonalizable if and only if every generalized eigenvector of

T

is an eigenvector of

T

. For

\mathbf{F} = \mathbf{C}

, this exercise adds another equivalence to the list of conditions for diagonalizability in 5.55.