I am preparing linear algebra exam and I met a problem about Jordan normal form.
Suppose $V$ is a vector space over field $\mathbb K$, $\psi$ is a linear transform on $V$, the char polynomial and minimal polynomial is defined as $f(\lambda)$ and $m(\lambda)$ respectly, and we have the irreducible decomposition as follows:
$$f(\lambda)=P_1(\lambda)^{r_1}P_2(\lambda)^{r_2}...P_3(\lambda)^{r_{t}}$$
$$m(\lambda)=P_1(\lambda)^{s_1}P_2(\lambda)^{s_2}...P_3(\lambda)^{s_{t}}$$
we define $\mu_i=dim \ker P_i(\lambda)^{r_{i}}$. Show that $r_i=\frac{\mu_i}{\deg p_{i}}$.
The problem is not hard but this really looks very similar to the definition of stability in some paper I used to contact. It follows from the stability of vector bundle defined as $\mu(E)=\frac{c_{1}(E)}{rank(E)}$ where $c_1(E)$ means the first Chern class . But I have to admit I still don't understand the motivation behind it...( I know it comes from Munford's $GIT$ but it was not my field, so I just accepted this concept at that time.)
I don't know does this really has some close connection (since Jordan normal form really leaves something invariant by irreducible factor, this really has some similar idea behind invariant theory I guess although you need some group action like $Gl_{n}{\mathbb (K)}$) or it was just a borrowed symbol. Hope someone can give some detailed background.