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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.

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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.

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