TITLE: The intuition behind generalized eigenvectors
QUESTION [29 upvotes]: An ordinary eigenvector can be viewed as a vector on which the operator acts by only stretching (without rotating) it.
Is there a similar intuition behind generalized eigenvectors?
EDIT: By generalized eigenvectors I'm referring to vectors in the kernel of $(T-\lambda I)^k$ for some $k$.

REPLY [2 votes]: Generalized eigenvectors also have an interpretation in system dynamics:If only generalized eigenvectors can be found the dynamics components (=blocks in Jordan canonical form) cannot be completely decoupled. Only if the respective matrix is fully diagonizable a full decoupling is possible.
See also this instructive video from Stanford university (from about 1:00 hours on):http://academicearth.org/courses/introduction-to-linear-dynamical-systems