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In one of my recent posts, I used the Jordan normal form for a matrix in order to justify a couple of arguments. As a student, I learned the derivation of this form twice: firstly (as an undergraduate) by using the minimal polynomial, and secondly (as a graduate) by using the structure theorem for finitely generated modules over a principal ideal domain. I found though that the former proof was too concrete and the latter proof too abstract, and so I never really got a good intuition on how the theorem really worked. So I went back and tried to synthesise a proof that I was happy with, by taking the best bits of both arguments that I knew. I ended up with something which wasn’t too different from the standard proofs (relying primarily on the (extended) Euclidean algorithm and the fundamental theorem of algebra), but seems to get at the heart of the matter fairly quickly, so I thought I’d put it up on this blog anyway.

Before we begin, though, let us recall what the Jordan normal form theorem is. For this post, I’ll take the perspective of abstract linear transformations rather than of concrete matrices. Let $T: V \to V$ be a linear transformation on a finite dimensional complex vector space V, with no preferred coordinate system. We are interested in asking what possible “kinds” of linear transformations V can support (more technically, we want to classify the conjugacy classes of $\hbox{End}(V)$, the ring of linear endomorphisms of V to itself). Here are some simple examples of linear transformations.

1. The right shift. Here, $V = {\Bbb R}^n$ is a standard vector space, and the right shift $U: V \to V$ is defined as $U(x_1,\ldots,x_n) = (0,x_1,\ldots,x_{n-1})$, thus all elements are shifted right by one position. (For instance, the 1-dimensional right shift is just the zero operator.)
2. The right shift plus a constant. Here we consider an operator $U + \lambda I$, where $U: V \to V$ is a right shift, I is the identity on V, and $\lambda \in {\Bbb C}$ is a complex number.
3. Direct sums. Given two linear transformations $T: V \to V$ and $S: W \to W$, we can form their direct sum $T \oplus S: V \oplus W \to V \oplus W$ by the formula $(T \oplus S)(v,w) := (Tv, Sw)$.

Our objective is then to prove the

Jordan normal form theorem. Every linear transformation $T: V \to V$ on a finite dimensional complex vector space V is similar to a direct sum of transformations, each of which is a right shift plus a constant.

(Of course, the same theorem also holds with left shifts instead of right shifts.)