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:


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*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.

*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).

*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.
 A: 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".


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*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...
A: 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.


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*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.

A: 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:


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*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.

*They are very useful in obtaining bounds on eigenvalues.

*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.

A: 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. 
A: 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.
