# 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

 $E_{\lambda}:=\{v\in V\mid(T-\lambda I)^{i}(v)=0\textrm{ for some positive % integer }i\}.$

Here are some properties of $E_{\lambda}$:

1. 1.

$W_{\lambda}\subseteq E_{\lambda}$, where $W_{\lambda}$ is the eigenspace of $T$ corresponding to $\lambda$.

2. 2.

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

3. 3.

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

4. 4.

$E_{\lambda_{1}}\cap E_{\lambda_{2}}=0$ iff $\lambda_{1}\neq\lambda_{2}$. 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. 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

 $V=\bigoplus_{\lambda\in S}E_{\lambda},$

where $S$ is the set of all eigenvalues of $T$.

6. 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,

 $[T]_{\beta}=\begin{pmatrix}J_{1}&O&\cdots&O\\ O&J_{2}&\cdots&O\\ \vdots&\vdots&\ddots&\vdots\\ O&O&\cdots&J_{n}\end{pmatrix}$

where each $J_{i}=[T|_{E_{\lambda_{i}}}]_{\beta_{i}}$ is a Jordan canonical form, and $O$ is a zero matrix.

7. 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 $v_{i} for $v_{i}\in\beta_{i}$ and $v_{j}\in\beta_{j}$ for $i, is an ordered basis for $V$ such that $[T]_{\beta}$ is a Jordan canonical form, being the direct sum of matrices $J_{i}$.

8. 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. Prentice-Hall Inc., 1997.
Title generalized eigenspace GeneralizedEigenspace 2013-03-22 17:23:36 2013-03-22 17:23:36 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 15A18 GeneralizedEigenvector