= Classification of six-dimensional nilpotent matrices by rank and nilpotency index
{title2=$(\operatorname{rank}A,m_A)\text{ determines similarity in size }6$}
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 are
$$
(6),\ (5,1),\ (4,2),\ (4,1,1),\ (3,3),\ (3,2,1),\ (3,1,1,1),\ (2,2,2),\ (2,2,1,1),\ (2,1,1,1,1),\ (1,1,1,1,1,1).
$$
No 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>.
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