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Is there any good reference for difficult problems in linear algebra? Because I keep running into easily stated linear algebra problems that I feel I should be able to solve, but don't see any obvious approach to get started.

Here's an example of the type of problem I am thinking of: Let $A, B$ be $n\times n$ matrices, set $C = AB-BA$, prove that if $AC=CA$ then $C$ is nilpotent. (I saw this one posed on the KGS Go Server)

Ideally, such a reference would also contain challenging problems (and techniques to solve them) about orthogonal matrices, unitary matrices, positive definiteness... hopefully, all harder than the one I wrote above.

edited Feb 12 2010 at 0:52

Grétar Amazeen

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This should be turned into a Community Wiki question. for there is no correct answer... – Mariano Suárez-Alvarez Feb 11 2010 at 23:42

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Here's another example perhaps: say A and B are real n x n matrices,and A^2+B^2=AB. If AB-BA is invertible, prove that n is a multiple of 3. Are there really books that can teach you how to solve such poroblems?? – Kevin Buzzard Feb 11 2010 at 23:43

How could I make this community wiki? – zeb Feb 12 2010 at 0:10

By editing the question and checking the box which says "Community Wiki". – Mariano Suárez-Alvarez Feb 12 2010 at 0:12

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@Darij: feel free to use any standard result you like from the representation theory of Lie algebras to resolve my question about real n x n matrices above! – Kevin Buzzard Feb 12 2010 at 14:56

show 5 more comments

Jonas Meyer · Answer 1 · 2010-02-11 23:48:05Z UTC

Halmos's Linear Algebra Problem Book. It contains problems, then hints, then solutions. There is a variety of difficulty levels, and some of the problems are very easy, but some are challenging. The book is designed to be a supplement for learning linear algebra by problem solving, so it may not have the focus you're looking for.

Wanderer · Answer 2 · 2010-02-11 23:38:07Z UTC

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Indeed, Halmos is a very good reference. You will also find some nice problems in Berkeley problems in mathematics and on the website of the International Mathematics Competition

answered Feb 11 2010 at 23:38

Wanderer

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Sorry to say but the Berkeley problems book is mostly spam. Most of the problems there are either well-known or boring. Actually, exam questions and problems which make one learn something are two totally different kinds of problems; the former are supposed to check one's skills, while the latter should develop them. As for the IMC, I agree - that contains many good questions, just as the [url=vjimc.osu.cz/index.php?page=history]Vojtech Jarnik[/url] and the [url=mat.itu.edu.tr/gungor/IMO/www.kalva.demon.co.uk/… contest[/url]. – darij grinberg Feb 12 2010 at 0:47

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I didn't say that all problems are interesting. But I found some of the problems in the book interesting. Not all of them are standard questions, using standard techniques...! – Wanderer Feb 12 2010 at 0:49

Mariano Suárez-Alvarez · Answer 3 · 2010-02-11 23:40:01Z UTC

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Google will find for you V. Prasolov's Problems and Theorems in Linear Algebra, which has beautiful more or less hard problems.

answered Feb 11 2010 at 23:40

Mariano Suárez-Alvarez

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Portland · Answer 4 · 2010-02-12 15:30:10Z UTC

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In addition to those mentioned above, there is Linear Algebra: Challenging Problems for Students by Fuzhen Zhang

answered Feb 12 2010 at 15:30

Portland

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Alekk · Answer 5 · 2010-02-12 16:49:27Z UTC

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you could also browse the linear algebra section of AoPS.

answered Feb 12 2010 at 16:49

Alekk

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Sunni · Answer 6 · 2010-03-04 02:09:03Z UTC

Let me introduce you a good reference, IMAGE. At the end of IMAGE there is a section called IMAGE Problem Corner: Solutions of Old Problems and New Problems. You may enjoy solving these problems and read solutions by others. See http://www.math.technion.ac.il/iic/IMAGE/

Gerhard Paseman · Answer 7 · 2012-03-03 20:10:41Z UTC

I will take this opportunity to post my favorite linear algebra problem. I call it 0 not equal to 1.

Let A be an nxn 0-1 matrix with nonzero determinant. Show that there is a 1 in every row and in every column of A, and further there is a permutation matrix P so that PA has a diagonal of all 1's.

Let B be an nxn 0-1 matrix with nonzero determinant. We cannot show that there is a 0 in every row and in every column, so assume B also has this property. Are there nxn permutation matrices P and Q such that PBQ has all 0's on the diagonal? If not, how small a trace can one guarantee?

Gerhard "Ask Me About System Design" Paseman, 2012.03.03

Federico Poloni · Answer 8 · 2012-03-03 22:16:29Z UTC

If you happen to know a little bit of Italian, another good resource is Problemi risolti di algebra lineare, by Broglia, Fortuna, Luminati.

(By the way, if you have never done it, reading a math book in another language is often easier than it seems at first sight)

Linear Algebra Problems?

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