Matroids over Hyperfields, Part II

In Part I of this post, we defined hyperrings and hyperfields, gave some key examples, and introduced matroids over (doubly distributive) hyperfields in terms of Grassman-Plücker functions.  If $E=\{ 1,\ldots,m \}$ is a finite set and $K$ is a field, we saw that a $K$-matroid on $E$ is the same thing as a linear subspace of $K^m$, and if ${\mathbb K}$ is the Krasner hyperfield then a ${\mathbb K}$-matroid on $E$ is the same thing as a matroid in the usual sense.  Matroids over the hyperfield ${\mathbb S}$ of signs are the same thing as oriented matroids, and matroids over the tropical hyperfield ${\mathbb T}$ are the same thing as valuated matroids.  In this post we will give some “cryptomorphic” axiomatizations of matroids over hyperfields, discuss duality theory, and then discuss the category of hyperrings in a bit more detail. Continue reading

The Jordan Canonical Form

In my current position as Director of Undergraduate Studies for the Georgia Tech School of Mathematics, I’ve been heavily involved with revamping our linear algebra curriculum. So I’ve spent a lot of time lately reading through various linear algebra books. The goal of this post is to give a self-contained proof of the existence and uniqueness of the Jordan Canonical Form which is somewhat different from the ‘usual’ proofs one finds in textbooks.  I’m not claiming any novelty — I’m sure this approach has been discovered before — but I don’t know a good reference so I thought I’d record the details here.

The proof I give here does not use properties of polynomials (e.g. the Chinese Remainder Theorem), nor does it rely on the classification of finitely generated modules over a PID, so it might be of some pedagogical interest. The proof I give for the Generalized Eigenvector Decomposition is based on an auxiliary result — the Fitting Decomposition — which in my opinion ought to be better known.  The proof I give of the structure theorem for nilpotent operators comes from these lecture notes of Curt McMullen (Theorem 5.19).  It is particularly concise compared to some other arguments I’ve seen. Continue reading