Every nilpotent matrix is similar to a sum of nilpotent Jordan blocks

**Theorem.** Every nilpotent matrix is similar to a matrix in Jordan form with blocks of the form:
$$J_k(0) = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & \cdots & 0 & 0 \end{pmatrix}$$

Each block has 1 on the superdiagonal and 0 everywhere else. The sizes of the blocks determine the similarity class.

**Example.** A $3 \times 3$ nilpotent matrix with index 2 is similar to:
$$\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$Copy