Nilpotent endomorphism/Jordan normal form/Fact

Let ${}K$ be a field and let ${}V$ denote a finite-dimensional ${}K$-vector space. Let

$\varphi \colon V\longrightarrow V$

be a nilpotent linear mapping.

Then there exists a

basis of ${}V$ such that describing matrix, with respect to this basis, has the form

${\begin{pmatrix}0&c_{1}&0&\cdots &\cdots &0\\0&0&c_{2}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&0&c_{n-2}&0\\0&\cdots &\cdots &0&0&c_{n-1}\\0&\cdots &\cdots &\cdots &0&0\end{pmatrix}},$

where ${}c_{i}$ equals ${}0$ or ${}1$.

That is, ${}\varphi$ can be brought into Jordan normal form.

Proof, Write another proof