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:
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.