Give an example of an operator T on a finitedimensional real

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.

Solution

Let T be the operator in R3 which sends

e1 to e2

e2 to -e1

e3 to 0

This has only eigenvalue as 0 for the eigenvector e3

But it is not nilpotent.