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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.)2012-02-28
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    The adjacency matrices of various directed graphs are interesting. Permutation matrices are special cases.2012-02-28
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    @Jonas: ??? What about AB-BA=[[0,1]|[1,0]]...2012-02-28
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    @Didier: Thanks for catching my error.2012-02-28
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    @Willie: Yes, that's what I was (not) thinking about, thanks!2012-02-28

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