Give an example of an operator $T \in L(\mathbb C^7,\mathbb C^7)$ such that $T^2+T+I$ is nilpotent.

Question

Give an example of an operator $T \in L(\mathbb C^7,\mathbb C^7)$ such that $T^2+T+I$ is nipotent.

Attempt:

If we choose $\lambda$ such that $\lambda^2+\lambda+1=0$ and define $T$ such that $T v = \lambda v~\forall~v \in \mathbb C^7$, then, $T^2+T+I$ is nilpotent for such a $T$.

Point of confusion A nilpotent operator should have $0$ as the only eigenvalue, but here it surely has others. Why is there this discrepancy?

Thank you for reading through!

Answer 1

The discrepancy is because $\lambda$ is an eigenvalue for $T$, not for $U := T^2 + T + I$ necessarily. In fact, $0$ is the only eigenvalue for $U$, since $U$ is the zero function.

Answer 2

If you do what you suggest then $T=\lambda I$ and $T^2+T+I$ is not only nilpotent, it's $0$. There is no problem with that, it's a valid solution.