generalized eigenspace
Let $V$ be a vector space^{} (over a field $k$), and $T$ a linear operator on $V$, and $\lambda $ an eigenvalue^{} of $T$. The set ${E}_{\lambda}$ of all generalized eigenvectors^{} of $T$ corresponding to $\lambda $, together with the zero vector $0$, is called the generalized eigenspace^{} of $T$ corresponding to $\lambda $. In short, the generalized eigenspace of $T$ corresponding to $\lambda $ is the set
$${E}_{\lambda}:=\{v\in V\mid {(T\lambda I)}^{i}(v)=0\text{for some positive integer}i\}.$$ 
Here are some properties of ${E}_{\lambda}$:

1.
${W}_{\lambda}\subseteq {E}_{\lambda}$, where ${W}_{\lambda}$ is the eigenspace^{} of $T$ corresponding to $\lambda $.
 2.

3.
If $V$ is finite dimensional, then $dim({E}_{\lambda})$ is the algebraic multiplicity of $\lambda $.

4.
${E}_{{\lambda}_{1}}\cap {E}_{{\lambda}_{2}}=0$ iff ${\lambda}_{1}\ne {\lambda}_{2}$. More generally, ${E}_{A}\cap {E}_{B}=0$ iff $A$ and $B$ are disjoint sets of eigenvalues of $T$, and ${E}_{A}$ (or ${E}_{B}$) is defined as the sum of all ${E}_{\lambda}$, where $\lambda \in A$ (or $B$).

5.
If $V$ is finite dimensional and $T$ is a linear operator on $V$ such that its characteristic polynomial^{} ${p}_{T}$ splits (over $k$), then
$$V=\underset{\lambda \in S}{\oplus}{E}_{\lambda},$$ where $S$ is the set of all eigenvalues of $T$.

6.
Assume that $T$ and $V$ have the same properties as in (5). By the Jordan canonical form theorem, there exists an ordered basis $\beta $ of $V$ such that ${[T]}_{\beta}$ is a Jordan canonical form. Furthermore, if we set ${\beta}_{i}=\beta \cap {E}_{{\lambda}_{i}}$, then ${[{T}_{{E}_{{\lambda}_{i}}}]}_{{\beta}_{i}}$, the matrix representation of ${T}_{{E}_{\lambda}}$, the restriction^{} of $T$ to ${E}_{{\lambda}_{i}}$, is a Jordan canonical form. In other words,
$${[T]}_{\beta}=\left(\begin{array}{cccc}\hfill {J}_{1}\hfill & \hfill O\hfill & \hfill \mathrm{\cdots}\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill {J}_{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill O\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {J}_{n}\hfill \end{array}\right)$$ where each ${J}_{i}={[{T}_{{E}_{{\lambda}_{i}}}]}_{{\beta}_{i}}$ is a Jordan canonical form, and $O$ is a zero matrix^{}.

7.
Conversely, for each ${E}_{{\lambda}_{i}}$, there exists an ordered basis ${\beta}_{i}$ for ${E}_{{\lambda}_{i}}$ such that ${J}_{i}:={[{T}_{{E}_{{\lambda}_{i}}}]}_{{\beta}_{i}}$ is a Jordan canonical form. As a result, $\beta :={\bigcup}_{i=1}^{n}{\beta}_{i}$ with linear order extending each ${\beta}_{i}$, such that $$ for ${v}_{i}\in {\beta}_{i}$ and ${v}_{j}\in {\beta}_{j}$ for $$, is an ordered basis for $V$ such that ${[T]}_{\beta}$ is a Jordan canonical form, being the direct sum of matrices ${J}_{i}$.

8.
Each ${J}_{i}$ above can be further decomposed into Jordan blocks, and it turns out that the number of Jordan blocks in each ${J}_{i}$ is the dimension^{} of ${W}_{{\lambda}_{i}}$, the eigenspace of $T$ corresponding to ${\lambda}_{i}$.
More to come…
References
 1 Friedberg, Insell, Spence. Linear Algebra. PrenticeHall Inc., 1997.
Title  generalized eigenspace 

Canonical name  GeneralizedEigenspace 
Date of creation  20130322 17:23:36 
Last modified on  20130322 17:23:36 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 15A18 
Related topic  GeneralizedEigenvector 