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. $E_{\lambda }$ is a subspace of $V$ and $E_{\lambda }$ is $T$-invariant.

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,

   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…