I read numerous demonstration of the existence of the Jordan Canonical Form, but all of them involve more than 2 pages of demonstration with numerous lemmas in between.
I'm writing some notes for some students, but the subject is only tangentially related to Jordan Normal Form and so I was wondering if anybody knew a simple 1-page demonstration of the existence of this form! If the demonstration is original I will find a way to cite you :)
$\newcommand{\span}{\operatorname{span}}$Are you assuming an algebraically closed closed ground field? If so then here is a proof. Let $W_{\lambda}=\{v :(T-\lambda)^kv=0, \text{for some $k$} \}$. Its not hard to show that $V=\bigoplus W_{\lambda}$, ill skip this but it does take some time. So look at the operator $N=T-\lambda$ in the space $W=W_{\lambda}$. $N$ is nilpotent. Chose $v_1, \ldots , v_p$ basis for $\ker N$, which has to be non zero. Now consider those $w$ such that $Nw\in \span\{ v_1\}$ these $w$ are one dimensional, modulo the $\ker N$ : since if there were $w$ and $w^{\prime}$ ,linearly independent, with a choice of constant $N(w-bw^{\prime})=0$ and so $w-bw^{\prime}\in \ker N$. So now chose some vectors such that $Nw_i \in \span\{ v_i\}$ where possible say $w_1\ldots w_q$ for $q\leq p$. Next chose $x_i$ such that $Nx_i\in \span\{ w_i\}$ and so on. Now your basis for $W$ is $v_1,w_1,x_1,y_1,\ldots v_2,w_2,x_2, \ldots v_3,w_3,x_3,\ldots$ its easy to see that under this basis $N$ and thus $T$ has the desired form on $W$. Its not 2 pages but it is also quite terse.