Classification of six-dimensional nilpotent matrices by rank and nilpotency index ($(\mathrm{rank}A,m_A)\text{ determines similarity in size }6$)

A complex nilpotent matrix of size six is determined up to matrix similarity by its matrix rank and nilpotency index. Its Nilpotent Jordan block sizes form an integer partition of six. If there are $k$ blocks and the largest size is $m$, then its matrix rank is $6-k$ and its minimal polynomial is $t^m$. The eleven possible size lists areNo two have the same pair $(m,k)$, which proves the assertion by Jordan normal form. The dimension restriction matters: in size seven, block lists $(3,3,1)$ and $(3,2,2)$ both have matrix rank four and minimal polynomial $t^3$, but give different Jordan normal forms.