Any suggestions for abstract algebra-multilinear algebra books? 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!
 A: 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.  
A: 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. 
A: 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.
