Permutation matrix/3-cycle/Invariant linear subspace/Minimal polynomial/Example

We consider the permutation matrix

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

The line ${}K{\begin{pmatrix}1\\1\\1\end{pmatrix}}$ is the eigenspace for the eigenvalue ${}1$. Moreover,

${}U=\langle {\begin{pmatrix}1\\-1\\0\end{pmatrix}},{\begin{pmatrix}0\\1\\-1\end{pmatrix}}\rangle \,$

is an invariant linear subspace (which, over ${}\mathbb {C}$, according to fact, can be decomposed further into smaller eigenspaces). With respect to the given basis, the restriction of the linear mapping to ${}U$ has the describing matrix

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

Therefore, the characteristic polynomial of this matrix is

${}X(X+1)+1=X^{2}+X+1\,.$

This is also the minimal polynomial of the restriction. The minimal polynomial of the permutation matrix is ${}X^{3}-1$, and indeed we have

${}X^{3}-1=(X-1)(X^{2}+X+1)\,$

in accordance with fact.