# Homework Help: Find eingenvalues and eigenvectors of an order n matrix

1. Jun 19, 2011

### elimax

1. The problem statement, all variables and given/known data

Being $T\in L(\mathbb{R}^n)$ a linear operador defined by $T(x_1, ... ,x_n )=(x_1+...+x_n,...,x_1+...+x_n )$, find all eigenvalues and eigenvectors of T.

2. Relevant equations

$det(T-\lambda I)=0, Ax=\lambda x$

3. The attempt at a solution

By checking n=1,2,3,4 I guess the answer is:

λ=n, x=(1,1,1)
λ=0 (multiplicity n-1), x such as , $\forall k \in \{1,...,(n-1)\}$, $x_k=1$, $x_n=-1$ and $x_i=0$ in all other positions. For instance, for n=4, we have (1,0,0,-1), (0,1,0,-1), (0,0,1,-1).

But how do I prove it for the general case? I'm trying induction, but I think I'm missing something...