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I want to read a little about these:

The characteristic polynomial and minimal polynomial of a $T \in\mathrm{End}(V)$, or given a matrix $A$, finding the Jordan form and when can I say it is diagonalizable.

Which books do you think it's good to read for these topics?

I have Hungerford's algebra book and I totally don't understand the multilinear algebra part. I'll need another book. Thank you!

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You might look at Roman's advanced linear algebra text, or perhaps Dummit and Foote's 3rd edition. These have all you want and a lot more plus some context beyond cut and dried linear. –  James S. Cook Dec 29 '12 at 5:35
    
thank you very much! –  treasure Dec 30 '12 at 2:43

3 Answers 3

Your question is more about linear algebra than multi-linear algebra. (The latter has the implication of tensor products and so on, whereas the only "multilinear" aspect of your question is the relationship to determinants, which is a standard linear algebra topic.)

Most texts on linear algebra will treat your question, and have exercises.

Axler's Linear algebra done right has a treatment that doesn't use determinants, which some people like. But there are dozens of books available that do use determinants as well; basically any book with linear algebra in the title will cover the topics you asked about, so just go to your school library and browse the linear algebra portion of the shelves to find a book that looks good for you.

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thank you very much! :p –  treasure Dec 30 '12 at 2:13

There aren't many multilinear algebra textbooks,even older ones. The wonderful text on algebra by E.B. Vinberg has a terrific chapter on it. Volume 1 of the treatise by Anthony Knapp has a very good chapter on it, more complete then Vinberg's.

As for actual whole textbooks, there are basically 3 of them: For multilinear algebra from a purely algebraic and formal point of view, there's the classic textbook by W.Grueb, Multilinear Algebra. which is very austere but comprehensive. Less difficult is the book of the same title by Northcott (a nearly forgotten algebraicist which I'd love to help republish the textbooks of one day). Lastly, there's a good discussion in the advanced linear algebra text of T.Y.Blyth, Module Theory.

As far as I know,that's all there is as far as "standard sources" go.

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Dear Mathemagician, The question is about char. polys and Jordan normal forms and char. polys, which are basic topics of linear algbera, which your answer doesn't really address, since there are dozens, if not hundreds, of texts available which treat linear algebra. Incidentally, there are many, many texts available that treat multilinear algebra as well, in addition to those that you mention. Regards, –  Matt E Dec 29 '12 at 4:31
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@Matt E. I thought it was rather obvious that there's a legion of textbooks on those subjects, so I decided to address the specific question of multilinear algebra, which there are NOT nearly as many sources available. –  Mathemagician1234 Dec 29 '12 at 5:09
    
Dear Mathemagician, Fair enough! Regards, –  Matt E Dec 29 '12 at 5:10
    
thank you guys. sorry that i didn't say it right. i mean linear algebra. i should have made it clear. –  treasure Dec 30 '12 at 2:16

I would recommend Linear Algebra Done Right by Sheldon Axler. The first 5 chapters should be a good revision, then you can jump to Chapters 8 and 9 to read about the characteristic polynomials and Jordan form.

The slight drawback is his unwillingness to talk about Determinants.

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That's kind of like doing King Kong without the ape, Calvin............. –  Mathemagician1234 Dec 29 '12 at 5:10
    
thank you so much for such detailed info. –  treasure Dec 30 '12 at 2:14

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