What can be said about the rank of Jordan blocks of size $n$? When is rank $J_n(\lambda) = n$?
From my observation, it seems that whenever $\lambda\neq 0$, the corresponding Jordan block has full rank (i.e. $n$). This seems clear from the fact that $J_n(\lambda)$ is upper triangular, and if we select any two columns $c_1,c_2$ and put $ac_1 + bc_2 = 0$, then we must have $a=b=0$. However this fails when $\lambda = 0$, and the matrix becomes strictly upper triangular.
I would appreciate any hints on how to be able to make generalized comments about the rank of Jordan blocks. Thank you!
I am adding this answer just for the sake of completeness, it is definitely trivial and unnecessary:
$$\text{rank }J_n(\lambda)= n-1 \text{ if }\lambda=0$$ and $$\text{rank }J_n(\lambda)=n \text{ if }\lambda\neq 0$$