# Topics for a matrix analysis course

I recently taught a new (to my department) course titled "Matrix Analysis". For various reasons that I won't go into here, I was dissatisfied with the textbook I (loosely) followed, and with every other textbook on the subject I've looked at. The next time I teach the class I will just follow my own notes, which I'm rewriting from scratch. Some of the vital characteristics of the class are the following:

1. It aims to be accessible and useful to a wide variety of students: grad students and advanced undergrads in pure and applied math, engineering grad students, and possibly others. Particular interests of faculty and grad students in my department which it aims to support include functional analysis, numerical analysis, and probability.

2. The prerequisite is one semester of linear algebra (although, with the point above in mind, I don't want to assume too much about exactly what that course includes).

3. As indicated by the title, the emphasis is on analytic aspects of linear algebra and matrix theory -- i.e., those involving convergence, continuity, and inequalities -- as opposed to more algebraic aspects.

Here's my question:

What topics do you think such a class should include, but might not?

The latter part of the question is just to exclude no-brainers like SVD and the Courant-Fischer min-max theorem. I'm especially looking for things that make you think, "Everyone should know about X. Why isn't it ever taught in classes?" Of course I already have in mind some topics of this sort, but by their very nature there are surely many other such topics I'd never think of on my own.

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I wrote a text book (Springer-Verlag GTM 216). Therefore I am potentially concerned by your statement I was dissatisfied with the textbook I followed.

Because my book is a bit advanced, I do not advise you to follow it. But let me extract a few topics that can be taught in the context you describe.

• The spectral radius. Matrix norms.
• The numerical radius, Toepliz-Hausdorff theorem.
• Non-negative matrices, Perron-Frobenius theorem.
• Matrix exponential, its use for ODEs.
• Positive definite Hermitian matrices, their square root, the polar decomposition.
• The Schur complement and Sherman-Morison formulae.
• Elementary methods for solving linear systems.
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Your book was not one of the ones I looked at, so there's no need for you to be concerned. (I also like the book I used very much as a reference, but found it ill-suited to teaching my class.) Thanks for the suggestions, and I'll definitely take a look at your book! –  Mark Meckes Apr 4 '11 at 14:26

Denis Serre's suggestions all sound very sensible. Here are a few which I think haven't yet been mentioned, and which might be at the right level even if they are "lower-priority" or "too specialized".

• Tridiagonalization of symmetric matrices (why not apply this to the GOE, ahem)

• Gershgorin's theorem

• Unitary similarity to an upper triangular matrix

However, I guess it depends on the type of linear algebra course you can assume the students have taken...

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–  Mark Meckes Apr 5 '11 at 11:45
Well, yes ;-) Trotter's paper in Adv Math is the earliest reference I've seen, but I have at best a superficial knowledge of the RM literature. –  Yemon Choi Apr 5 '11 at 19:12
Well,just speaking for myself,i first saw Gershgorin's theorem in my second semester of linear algebra.I think any matrix analysis course that doesn't have a full discussion of the Jordan form and its role in linear analysis is seriously missing the point. –  Andrew L Apr 9 '11 at 23:26
Thanks for the info, Andrew. I can't remember where I was first shown Gershgorin's theorem -- it may be one of those topics which is not left out of 2nd courses because of difficulty, but just for reasons of time. –  Yemon Choi Apr 10 '11 at 5:32
@Andrew L: Indeed reasons of time force many desirable topics to be omitted. I haven't decided about Gershgorin's theorem, because I'm not sure whether it's useful enough to favor it over other topics. I almost certainly will omit Jordan form. Although Jordan form is useful for understanding the algebra of matrices, it is badly behaved from an analytic point of view, and so it doesn't really fit into this class. –  Mark Meckes Apr 14 '11 at 15:22

I'd actually urge you to reconsider your stance on the Courant-Fischer theorem. I think it's very fitting for your course for at least 3 reasons:

1. It shows that the eigenvalues are "there for a reason" - they are not just random numbers who happen by luck to solve the eigenequation, they solve optimization problems that make sense. Now, you may want to prove just the extremal characterizations - they are much easier - and to state the more compicated general for without proof.
2. They are very useful in obtaining bounds on eigenvalues.
3. They keep popping up in applications. To say nothing of the physical problems that motivated the Rayleigh form in the first place, it comes up in spectral clustering and in the derivation of the Fisher linear discriminant - two applications that most engineering students will be sure to appreciate.
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I think you may have misread his statement. –  Chris Godsil May 2 '12 at 13:13
I think you misunderstood my comment. What I meant is that obviously any such course has to include the Courant-Fischer theorem; I'm interested in less obvious suggestions. –  Mark Meckes May 2 '12 at 13:13
Yes, of course, silly me. I was writing (and reading, it turns out) in great haste. Sorry. But I think the points I tried to make are valid nevertheless. –  Felix Goldberg May 2 '12 at 17:28

I like Richard Bellman's book Introduction to Matrix Analysis and its exercises. Maybe stating the reasons why you didn't like the books you looked at may help others suggest ones more adapted to your taste. That said, Denis Serre's list of topics looks fine. I would just add, although this is not really orthodox, Dunford functional calculus and Markov chains as topics.

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I haven't looked at Bellman's book; thanks for the suggestion. The short version of why I didn't like other books is that they don't fit the three criteria I listed very well. For example, to mention my two favorite references on the subject, Horn and Johnson's emphasis of topics is rather different from what I want, and Bhatia assumes too many prerequisites. Functional calculus and Markov chains are definitely among the often-omitted topics I like to cover. –  Mark Meckes May 2 '12 at 13:18
I did my dissertation on a problem in Matrix Analysis/Operator Theory and found Bhatia's book to be helpful. If the students are familiar with the usual $L_p$ –  Mustafa Said Sep 4 '13 at 14:07

I wrote my dissertation on a problem in matrix analysis and I found that I had to read from several different sources to understand the material. I don't know which is the best book on the subject but I would suggest reading Bhatia's book and Horn's book. In addition to this, I would suggest that the students read a short research paper such as "Almost Commuting Unitaries" by Exel and Loring. The paper is only three pages and only assumes knowledge of basic real analysis/complex analysis and basic linear algebra. Moreover, it gives the class a good example of the interplay between analysis and linear algebra.

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