# Generalized Eigenvectors and Eigenspaces

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.

Diagonalization is such a convenient property of matrices, but not every matrix can be diagonalized. For example this matrix.

$\begin{bmatrix} 1&1\\0&1\end{bmatrix}$

This matrix is in "Jordan Normal" form. And every matrix can be put into Jordan Normal form.

$A = P J P^{-1}$

And the generalized eigenvectors form the columns of P.

And while not quite as easy to work with as a diagonal matrix, it is better than nothing.