- For a complex square matrix $M$, a maximal set of linearly independent eigenvectors for an eigenvalue $\lambda$ is determined by solving $ (M - \lambda I) x = 0. $ for a basis in the solution subspace directly as a homogeneous linear system.
- For a complex square matrix $M$, a generalized eigenvector for an eigenvalue $\lambda$ with algebraic multiplicity $c$ is defined as a vector $u$ s.t. $ (M - \lambda I)^c u = 0. $ I wonder if a generalized eigenbasis in Jordan decomposition is also determined by finding a basis in the solution subspace of $(M - \lambda I)^c u = 0$ directly in the same way as for an eigenbasis? Or it is more difficult to solve directly as a homogeneous linear system, and some tricks are helpful?
Thanks!