Generalised eigenvectors for Jordan chains

Question

Supposing we have a matrix $B$ whose characteristic polynomial is $(x-1)^4$ and $(B-I)^2=0$ holds true. How do I calculate the generalised eigenvalues of degree 2? For example take: $$ \begin{matrix} 1 & 0 & 2 & -1 \\ 0 & 1 & -2 & 1 \\ 1 & 1 & 1 & 0 \\ 2 & 2 & 0 & 1 \end{matrix} $$

Answer 1

You may calculate $Z_1=$ ker $(B-I)$ (you find 2 vectors $u_1$ and $u_2$). Then pick arbitrarily two independent vectors in ker$(B-I)^2$ (which means any two vectors in the complement of $Z_1$) and find linear combinations that maps to $u_1$ and $u_2$, respectively. (Their existence is guaranteed by dim $Z_1$=2 and $(B-I)^2=0$).