Consider this problem:

*Let $\pi$ be a permutation of integers $\{1,2,...,n\}$. If $x = (x_1, x_2, ...,x_n)$ is a vector in $\mathbb{C}^n$, write $Ax = (x_{\pi(1)},x_{\pi(2)},...,x_{\pi(n)})$. Find the spectrum of A.*

My thoughts:

1) Since $A$ is a permutation of $x$, one of these permutations must be the identity permutation, for which the eigenvalue should be $1$.

2) The only other eigenvalue should be $0$?? Since there shouldn't exist another $\lambda$ for which $Ax = \lambda x$ since the direction of $x$ isn't preserve in all other permutations?

Are my thoughts correct? What **is** the spectrum of $A$? And what would $A$ look like?