Why $J_k(0)^T$ ~ $J_k(0)$?

Question

Why these two Jordan blocks are similar: $J_k(0)^T $ and $J_k(0)$ ?

Answer 1

Because if $k_1+k_2+\dots + k_s=k$, all $k_i\geqslant 1$, $s>1$ then $$ \left(\bigoplus_{i=1}^{s}J_{k_i}(0)\right)^{k-1}=O $$ whereas $$ J_k(0)^{k-1}\not=O. $$