# Timing of when Cayley-Hamilton theorem is taught in Linear Algebra

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.

• 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." – Ben Crowell May 19 '18 at 20:45
• Out of interest, what do you mean by “the computational proof” and “the abstract proof”? – DavidButlerUofA May 19 '18 at 22:35
• @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. – Sheri A. May 20 '18 at 16:20
• @SheriA. Thank you. Perhaps adding a couple of phrases to clarify these would make your question easier to understand. – DavidButlerUofA May 20 '18 at 19:38