= Kernel dimensions of powers of a nilpotent map
{title2=$d_i=\dim\ker T^i$}
= Nullity sequence of a nilpotent map
{synonym}
= Nilpotent nullity sequence
{synonym}
For a <nilpotent linear map> with <minimal polynomial> $x^m$, set $K_i=\ker T^i$ and $d_i=\dim K_i$, with $d_0=0$. Then
$$
\boxed{0=d_0<d_1<\cdots<d_m=\dim V,\qquad d_{i+1}-d_i\leq d_i-d_{i-1}}.
$$
For strictness, choose $w$ with $T^{m-1}w\ne0$. The <vector> $T^{m-1-i}w$ belongs to $K_{i+1}\setminus K_i$ for $0\leq i<m$. For the second inequality, the <rank-nullity theorem> gives
$$
\dim T^i(K_{i+1})=d_{i+1}-d_i,\qquad
\dim T^{i-1}(K_i)=d_i-d_{i-1}.
$$
The first image is contained in the second because $T(K_{i+1})\subseteq K_i$.
If the <Nilpotent Jordan block> sizes are $s_1,\ldots,s_r$, then
$$
d_i=\sum_{j=1}^r\min(i,s_j),\qquad d_i-d_{i-1}=\#\{j:s_j\geq i\}.
$$
Thus successive increments count the <Jordan blocks> reaching a given size, and $(d_i-d_{i-1})-(d_{i+1}-d_i)$ counts blocks of size exactly $i$, with $d_{m+1}=d_m$. The <nullities> determine the entire <Jordan normal form>. Conversely any positive nonincreasing integer increments summing to $\dim V$ are realized by taking the corresponding numbers of each <Nilpotent Jordan block>.
Back to article page
