Finding eigenvalues. I'm working on the following problem:
Define $T \in L(F^n)$ (T an operator) by
$T(x_1,...,x_n) = (x_1+...+x_n,...,x_1+...+x_n)$
Find all eigenvalues and eigenvectors of $T$.
I've found that the eigenvalues of $T$ are $\lambda = 0$ and $\lambda = n$. Is there an easy way to prove that these are the only eigenvalues of $T$? Determining and solving the characteristic polynomial is messy for arbitrary $n$.

Try this:  by direct computation,
$T^2 = nT, \tag{1}$
since every entry of $T^2$ is $n$.
So $m_T(x) = x^2 - nx$ is the minimal polynomial of $T$; every eigenvalue $\lambda$ of $T$ satisfies
$m_T(\lambda) = 0, \tag{2}$
so the only possibilities are $\lambda = 0$ and $\lambda = n$.
