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elimax

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## Homework Statement

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

## Homework Equations

[itex]det(T-\lambda I)=0, Ax=\lambda x[/itex]

## 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 , [itex]\forall k \in \{1,...,(n-1)\}[/itex], [itex]x_k=1[/itex], [itex]x_n=-1[/itex] and [itex]x_i=0[/itex] 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...

Thanks in advance! :)

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