Minimal and characteristic polynomial of $\phi$

Question

Let V be a vector space over a field F with a basis (e1, . . . , en), and the operator

$\phi : e_i = \left\{ \begin{array}{ll} e_{i+1} & \mbox{if } i\ne j \\ e_1 & \mbox{if } i=j \end{array} \right.$

I want to show that the minimal polynomial of ϕ and its characteristic polynomial are both equal to $t^n − 1$. First at all I write the matrix of phi which is :

$\begin{pmatrix} 0 & 0 & ...&...& 0 &1\\ 1 & 0&&&&.\\ 0 &1&0&&&. \\ ... & ... &...&...&...&.\\ 0 &0&0&...&1&0\\ \end{pmatrix}$

The I want to compute the characteristic polynomial which is :

det(t-$\phi$) = det$\begin{pmatrix} t& 0 & ...&...& 0 &1\\ 1 & t&&&\\ 0 &1&t& \\ ... & ... &...&...&...&...\\ 0 &0&0&...&t\\ 0 &0&0&...&1&t\\ \end{pmatrix}$

What can I do now?

Answer 1

It's relatively easy to see that $\phi$ shifts the coordinates one place cyclically, and only after $n$ applications will they be in their original positions. Hence we immediately get the characteristic and minimal polynomials as $t^n-1$.

Indeed the given matrix is a companion matrix for $t^n-1$, so it is its own rational canonical form consisting of one invariant factor.