Definition: If $$A$$ is an $$n × n$$ matrix, a generalized eigenvector of $$A$$ corresponding to the eigenvalue $$λ$$ is a nonzero vector $$\mathbf X$$ satisfying $$(A − λI)^p ~\mathbf X = \mathbf 0$$ for some positive integer $$p$$.
Equivalently, it is a nonzero element of the nullspace of $$~(A − λI)^p~.$$
• A regular eigenvector is a generalized eigenvector of order $$1$$.
• Every $$n × n$$ matrix $$A$$ has $$n$$ linearly independent generalized eigenvectors associated with it.