I teach at a primarily undergraduate 4-year college in the US and we don't cover the proof of Cayley-Hamilton theorem in our linear algebra courses. I did however see both the computational and the abstract proof of the theorem when I studied from Hoffman-Kunze in my undergraduate (long time ago). I am wondering if anyone can share at what point in their institution's linear algebra courses the theorem is usually taught and at what level. Is it after eigenvalues/eigenvectors? Is it right after determinants are introduced? Is it taught in its most generality, such as "T is a linear operator over a vector field V with scalars k" or is it taught in a specialized case with linear transformations over C^n? I'm mostly interested to hear from people teaching at 4-year institutions in the US, but if the topic is taught at a 2-year college linear algebra course in the US, I would like to hear that too. Thank you for any information.
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1$\begingroup$ As one data-point, I double-majored in math and physics at a university in the US, then went on to get a PhD in physics. I had never heard of the Cayley-Hamilton theorem until today. I still have my sophomore linear algebra text, so I looked in the index, and I found that the theorem is stated and proved briefly in an appendix on optional topics, but there are no applications or interpretation. It seems like something that is in there in order to get adoptions of the text, so that the publisher can tell professors who want the theorem in the book, "Yes, it's in the book." $\endgroup$– user507Commented May 19, 2018 at 20:45
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$\begingroup$ Out of interest, what do you mean by “the computational proof” and “the abstract proof”? $\endgroup$– DavidButlerUofACommented May 19, 2018 at 22:35
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$\begingroup$ @DavidButlerUofA I'm thinking of the computational proof as something similar to the proof in the Wikipedia page, or something which uses the adjugate for expressing the determinant (which is what Hoffman-Kunze uses as the first proof). The abstract proof in Hoffman-Kunze came after the cyclic decomposition of the vector space in the context of discussing rational forms. So I'd consider something which refers to the vector space structure as abstract. $\endgroup$– Sheri A.Commented May 20, 2018 at 16:20
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$\begingroup$ @SheriA. Thank you. Perhaps adding a couple of phrases to clarify these would make your question easier to understand. $\endgroup$– DavidButlerUofACommented May 20, 2018 at 19:38
2 Answers
I see no compelling reason to teach Cayley-Hamilton in a first course directed at students who are not studying math as such. For math majors at a typical institution it probably makes more sense to teach Cayley-Hamilton in a first course on commutative algebra, rather than in a first-course on linear algebra. The principal use of Cayley-Hamilton in such courses is probably in the context of relating the minimal and characteristic polynomials and the deduction of canonical forms other than the Jordan form, and none of that material is typically included in the standard introductory linear algebra course. The theorem is not easy to state carefully (so that it does not appear to be a tautology) and the abstraction required to prove it is above the level of such a course. If one feels the need to indicate it, one can state it, and give exercises/examples checking it for particular linear transformations/matrices.
For what it's worth, I teach in Spain, in an engineering school. Our linear algebra class is taught in the first-year and has a higher level (more formal, more rigor) than what is standard in the sort of of US institution you ask about, but we do not teach Cayley-Hamilton.
In the US, Cayley-Hamilton isn't typically taught in a first course in Linear Algebra (non proof-based, intended for students majoring in the sciences and engineering), but is typically taught in a second more advanced course for mathematics majors. You can see this by looking at textbooks intended for these two different kinds of linear algebra courses. The textbook by Hoffman and Kunze that you mention is very commonly used in more advanced courses for mathematics majors.
The recommendations of the Linear Algebra Curriculum Study Group have had a major influence on the content of the first course in linear algebra. See
Carlson, David, Charles R. Johnson, David C. Lay, and A. Duane Porter. "The Linear Algebra Curriculum Study Group recommendations for the first course in linear algebra." The College Mathematics Journal 24, no. 1 (1993): 41-46.
