I have been given the following task:
Let $T\in\mathbb{M}_n(\mathbb{R})$ be an idempotent.
1) What are the possible eigenvalues of $T$?
2) Find examples in $\mathbb{M}_n(\mathbb{R})$ showing that the potential eigenvalues can be realized.
My answer to the first question:
A linear map, $T\in End(V)$ is called idempotent if $T^2=T$. then: $$ Tv=\lambda v\Longrightarrow T^2v=\lambda Tv=\lambda^2v$$ And using the condition for an idempotent we then have: $$T^2 v=Tv=\lambda^2v\Longrightarrow\lambda=\lambda^2$$ And it follows that the eigenvalues can be either $0$ or $1$.
Now for the second question, I do not know what the notation $\mathbb{M}_n(\mathbb{R})$ means, and therefore do not know how to tackle that question.