TITLE: Showing when a permutation matrix is diagonizable over $\mathbb R$ and over $\mathbb C$
QUESTION [7 upvotes]: For a permutation $\sigma$ of the set $\{1,...,n\}$, and consider the $n \times n$ matrix $A_\sigma$, where the $i^{\text{th}}$ column is the standard vector $e_{\sigma (i)}$.  For which $\sigma$ is $A_\sigma$ digonizable over $\mathbb{C}$, and which values for $\mathbb{R}$?
I don't exactly know what I should do.  I think I should use the fact that a map is diagnizable iff the characteristic polynomial of t factors in linear factors over the field I'm in, and if $m_g(\lambda)=m_a(\lambda)$ for all eigenvalues 
If anyone could start me up or help me out or do one of the examples it would be appreciated, thanks!
Small edit: I don't even know what the matrix looks like, maybe it's just me but I don't find it's clear

REPLY [8 votes]: This adresses the question of diagonalisability over $\Bbb R$. Over $\Bbb C$, there is no problem, since $A_\sigma^m=I_n$ where $m$ is the order of $\sigma$ in the group $S_n$, and thus $A_\sigma$ is diagonalizable over $\mathbb C$.

Use the cycle decomposition of $\sigma$ : if $\sigma=c_1\cdots c_p$ where the $c_i$ are cycles with disjoint support, then you can rearrange the standard basis $(e_1,\dots,e_n)$ so that the supports of the $c_i$ are "together and in order". This shows that $A_\sigma$ is conjugate to a matrix like so
$$\begin{pmatrix}C_1\\&C_2\\ &&\ddots\\&&& C_p\end{pmatrix}$$
where each $C_i$ is the matrix
$$C_i\;=\begin{pmatrix}
0&&&&&1\\
1&0\\
&1\\
&&&\ddots&\\
\\
&&&&1&0\end{pmatrix}_{l_i\times l_i}$$
where $l_i$ is the length of the cycle $c_i$. This matrix has minimal polynomial $X^{l_i}-1$.
Thus, the minimal polynomial of $A_\sigma$ is
$$\mu_{A_\sigma}=\mathrm{lcm}\,(X^{l_1}-1,X^{l_2}-1,\dots,X^{l_p}-1)$$
Since $X^n-1$ has non real roots for all $n\geq 3$, $A_\sigma$ is diagonalizable over $\Bbb R$ iff $\sigma$ is a (possibly empty) product of disjoint transpositions, i.e. $$A_\sigma \text{ is diagonalizable over } \Bbb R \Longleftrightarrow\sigma^2=\mathrm{id}$$