Cool/Useful Examples of Characteristic and Minimal Polynomials?

I'm teaching a Linear Algebra II undergrad course and for the section on characteristic & minimal polynomials, I really don't want to just give the students a bunch of matrices that have no meaning and ask them to find the char/min poly. I'm looking for cool/useful examples. Got any favourites?

So far, the only cool/useful examples I can think of are the characteristic poly of a companion matrix (since companion matrices will come up in other math courses the students might take) and the char&min polys of matrices of the form a's on the diagonal and b's everywhere else (yes, this is "cool" in my opinion, because once you do the general case, you can just read off the answer for a specific matrix of this form).

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How about the characteristic polynomial of left multiplication by an algebraic number $\alpha$ on $\mathbb{Q}(\alpha)$? (This is secretly just a companion matrix in a suitable basis but I think it's a good exercise to get students to notice this.) – Qiaochu Yuan Feb 28 '12 at 3:54
The adjacency matrices of various directed graphs are interesting. Permutation matrices are special cases. – Hans Engler Feb 28 '12 at 4:14
@Jonas: ??? What about AB-BA=[[0,1]|[1,0]]... – Did Feb 28 '12 at 8:10
@Didier: Thanks for catching my error. – Jonas Meyer Feb 28 '12 at 15:40
@Willie: Yes, that's what I was (not) thinking about, thanks! – Jonas Meyer Feb 28 '12 at 15:40

You can use either the minimal or characteristic polynomial $p(z)$ of $A$ to find $(A - cI)^{-1}$ for any scalar $c$ that is not an eigenvalue of $A$: expand out
$p(t+c) = \sum_{j=0}^m a_j t^j$ so $p(z) = \sum_{j=0}^m a_j (z - c)^j$, note that $a_0 = p(c) \ne 0$, and then $(A - cI)^{-1} = - \sum_{j=1}^m \frac{a_j}{a_0} (A - cI)^{j-1}$

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I'm not always that good a judge of "coolness", but I think the Clement-Kac(-Sylvester) matrices might be a teeny bit interesting. They are $n\times n$ unsymmetric tridiagonal matrices that take the form

$$\begin{pmatrix}0&1&&&\\n&0&2&&\\&n-1&\ddots&\ddots&\\&&\ddots&0&n\\&&&1&0\end{pmatrix}$$

which has positive and negative integer eigenvalues.

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It may be a little too advanced but I find the use of this stuff on elliptic curves fascinating. Counting points mod $p$ on an elliptic curve turns out to be the same as plugging 1 into the characteristic polynomial of a certain linear map on a 2-dimensional space.

The even nicer thing is that once you have done this you can determine the number of points over ANY finite field of characteristic p. This boils down to the fact that "the eigenvalues of the $n$th power of such a linear map are the $n$th powers of the eigenvalues of the linear map".

(Of course all of this can be restated in terms of matrices).

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The answer to this question Roots with equal fractional parts use an elegant argument with minimal polynomials.

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