Let $V$ be a finite-dimensional inner product space, and assume given a self-adjoint linear transformation $T:V\to V$ such that $T^3 = T$.
How do I show that the only possible eigenvalues of $T$ are $\lambda = 0, 1,$ or $−1$? And how do I write the corresponding eigenspaces?
That is what I did so far:
We have $T^3 = T$.
So if $\lambda$ is an eigenvalue of $T$ then we have $\lambda^3 = \lambda$.
$\lambda^3 = \lambda$
$\lambda(\lambda^2 - 1) = 0$
so $λ = 0$ or $λ =1$ 0r $λ = -1$.
Now, how do we write the eigenvectors?