This question is related to Jordan Forms. Suppose we are converting a $5*5$ matrix $A$ into Jordan form and that we have found $3$ basis vectors in $\ker (A - \lambda I)$. When we are finding $\ker((A - \lambda I)^2)$ then we found its nullity to be, say, $4$. That means $3$ vectors found before and one additional vector form this new kernel. How can we find this independent vector, or is it just by observation always?
Similarly if nullity would have been 5, then how to methodically find the 2 new independent vectors?
Thanks