Show $φ^n= 0$ for $φ:V→V$ a linear endomorphism such that $\mathrm{tr}(\wedge^qφ) = 0$ and $\dim(V) = n$ a vector space.

Question

As it is said in the title I'm working on an $n$-dimensional vector space $V$. My goal is to show that a linear endomorphism $\varphi:V \to V$ with $\mathrm{tr}(\wedge^qφ) = 0$ for all $q \geq 1$ is such that $φ^n= 0$.

From what I understand I need to show that $\varphi$ is a nilpotent operator, which I've never done before. I already found some answers on the wikipedia page https://en.wikipedia.org/wiki/Engel%27s_theorem and on the mathstack page Endomorphism- Nilpotent matrices, but since I am not sure to catch well the meaning of Engel's theorem and I didn't see it in class, I was wondering if there is any other way to prove this statement? Any idea ?

Answer 1

This is a direct consequence of Cayley-Hamilton Theorem. See this post Proving that the coefficients of the characteristic polynomial are the traces of the exterior powers

The given condition and the above post shows that the characteristic polynomial of $\varphi$ is $T^n$. Cayley-Hamilton shows $\varphi^n=0$.