Linear Algebra Done Right · Problem 8A.23

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · nilpotent-operators

Problem

Give an example of an operator

T

on a finite-dimensional real vector space such that

0

is the only eigenvalue of

T

but

T

is not nilpotent. This exercise shows that the implication (b)

\Rightarrow

(a) in 8.17 does not hold without the hypothesis that

\mathbf{F} = \mathbf{C}