Let $P_{\sigma}$ be a the permutation matrix of a cyclic Permutation $\sigma \in S_n$, i.e. $\sigma(n) = 1$ and $\sigma(i) = i+1$ for all $1 \leq i < n$.
My solutions sheet says that the characteristic polynomial of $P_\sigma$ is $\textrm{char}_{P_\sigma}(X) = X^n – 1$.
Now my problem is that if I try to calculate it I only get this solution in the case that n is even. If n was odd I get that $\textrm{char}_{P_\sigma}(X) = 1 – X^n$ which is very similar and would have the same eigenvalues but not exactly the same and this really confuses me (the exercise was more concerned with finding the eigenvalues of $P_\sigma$). Did I make a mistake with my calculations or are the solutions just not quite right?
For example let $\sigma \in S_3$ be a cyclic Permutation and thus
$$P_\sigma=\begin{pmatrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 &0 \end{pmatrix}. $$
Then I would calculate
$$\textrm{char}_{P_\sigma}(X) = \det(P_\sigma – X \cdot E_n) =\begin{pmatrix}-X & 0 & 1 \\ 1 & -X & 0 \\ 0 & 1 &-X \end{pmatrix} $$
and by Laplace expansion along the last column I would get
$$\textrm{char}_{P_\sigma}(X) = 1 – X^n.$$
Where is the problem?
Thanks for any help!
PS: This generally got me thinking about the concept of a characteristic polynomial. In class it was only introduced with respect to the calculation of eigenvalues which are the zeros of the characteristic polynomial so one could argue that we only really care about the zeros of this function. But now I was thinking
whether the function itself (not only looking at its zeros) was of any use and if so how? I couldn't find an answer yet despite searching for it.
Best Answer
The important defining feature of a characteristic polynomial is the nature of its roots.
The equations $1-x^n=0$ and $x^n-1=0$ have the same roots and I was taught to consider both $1-x^n$ and $x^n-1$ as the characteristic polynomial.
I presume your textbook simply writes them in descending powers of $x$.
Re. your other question the Cayley-Hamilton Theorem shows the importance of the characteristic polynomial as a polynomial satisfied by the transformation itself. It is of considerable value when determining the minimal polynomial satisfied by the transformation and thus further properties of the transformation.