Cyclic element in generalized eigenspace Generalized eigenspaces$ H_\lambda$ are a direct sum of cyclic vector spaces $V_1,...,V_n$, which means that there exists an element such that for a given endomorphism $\{v,Av,...,A^{j-1}v\}$ is a basis of such a $V_1,...,V_n$. Now I was wondering how one could find this element for all $V_1,...,V_n$ in a generalized eigenspace? 
 A: Once you have finished formulating the question correctly, you will no doubt find that you are asking for constructing a decomposition of the generalised eigenspace into Jordan blocks. This decomposition is not unique in general, and a general description of how it can be found can be messy. But after restricting to the generalised eigenspace for one eigenvalue $\lambda$, and subtracting $\lambda I$ from the endomorphism so that what remains is a nilpotent endomorphism$~N$, the following is a possible procedure.
Find the smallest power $k$ such that $N^{k-1}\neq0$, and a vector$~v_1$ that is not killed by$~N^{k-1}$. Choose a linear form $\alpha:V\to\Bbb C$ that does not vanish on $N^{k-1}(v_1)$. Then the space $V_1=\langle v_1,N(v_1),\ldots,N^{k-1}(v_1)\rangle$ can be the first Jordan block, and $C=\{\,v\in V \mid \alpha(v)=0, \ \alpha(N(v))=0, \ \ldots, \ \alpha(N^{k-1}(v))=0\,\}$ is an $N$-stable complement of $V_1$ in$~V$. Now continue with $C$ in place of $V$, using induction on the dimension.
