difference between eigenspace and generalized eigenspace What are the differences between eigenspace and generalized eigenspace? Why do we need generalized eigenspace? Can an arbitrary matrix (not necessarily over $\mathbb{C}$) have a Jordan form? Thank you very much.
 A: *

*The eigenspace of (a square matrix) $A$ corresponding to $\lambda$ is the collection of all vectors $\mathbf{x}$ that satisfy $A\mathbf{x}=\lambda\mathbf{x}$, or equivalently, $(A-\lambda I)\mathbf{x}=\mathbf{0}$. The generalized eigenspace of $A$ corresponding to $\lambda$ is the collection of all vectors $\mathbf{x}$ for which there exists a positive integer $k$ for which $(A-\lambda I)^k\mathbf{x}=\mathbf{0}$. The former is contained in the latter, but need not be equal. For example, with
$$A = \left(\begin{array}{cc}1&1\\0&1\end{array}\right),$$
and $\lambda=1$, you can check easily that the eigenspace consists only of the vectors of the form $(x,0)$ for some arbitrary $x$; whereas the generalized eigenspace is the larger collection of all vectors $(x,y)$, with $x$ and $y$ both arbitrary.

*We "need" generalized eigenspaces because the eigenspaces in general do not suffice to describe the entire space (the generalized eigenspaces may not suffice either). 

*A matrix $A$ with coefficients in a field $F$ has a Jordan canonical form (over $F$) if and only if the characteristic polynomial of $A$ splits over $F$.  In particular, all matrices over $F$ have a Jordan canonical form over $F$ if and only if $F$ is algebraically closed. But it's certainly possible to have specific matrices, say over $\mathbb{R}$, that have a Jordan canonical form. 
