I'm having trouble wrapping my head around these concepts. I know that the formal definition of a generalized eigenvector corresponding to $\lambda$ is a nonzero vector $x \in V$ s.t. $(T - \lambda I)^p (x) = 0$, for some positive integer $p$, and a generalized eigenspace corresponding to $\lambda$ is the subset of $V$ containing all gen. eigenvectors corresponding to $\lambda$.
The issue for me is that while I can understand what these definitions are saying mathematically, I can't understand them intuitively. What does it really mean to "generalize" an eigenvector outside of the mathematical definition? Any kind of "dumbed down" explanation would be tremendously helpful. Thank you.