PlanetMath: generalized eigenspace

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (21)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

generalized eigenspace

(Definition)

Let be a vector space (over a field ), and a linear operator on , and $\lambda$ an eigenvalue of . The set $E_{\lambda}$ of all generalized eigenvectors of corresponding to $\lambda$ , together with the zero vector 0, is called the generalized eigenspace of corresponding to $\lambda$ . In short, the generalized eigenspace of corresponding to $\lambda$ is the set

$% latex2html id marker 419 $\displaystyle E_{\lambda}:=\lbrace v\in V\mid (T-\lambda I)^i(v)=0\textrm{ for some positive integer }i\rbrace.$$

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

$W_{\lambda}\subseteq E_{\lambda}$ , where $W_{\lambda}$ is the eigenspace of corresponding to $\lambda$ .
$E_{\lambda}$ is a subspace of and $E_{\lambda}$ is -invariant.
If is finite dimensional, then $\dim(E_{\lambda})$ is the algebraic multiplicity of $\lambda$ .
$E_{\lambda_1}\cap E_{\lambda_2}=0$ iff $\lambda_1\ne \lambda_2$ . More generally, $E_A\cap E_B=0$ iff and are disjoint sets of eigenvalues of , and (or ) is defined as the sum of all $E_{\lambda}$ , where $\lambda\in A$ (or ).
If is finite dimensional and is a linear operator on such that its characteristic polynomial splits (over ), then
$\displaystyle V=\bigoplus_{\lambda\in S} E_{\lambda},$
where is the set of all eigenvalues of .
Assume that and have the same properties as in (5). By the Jordan canonical form theorem, there exists an ordered basis $\beta$ of such that $[T]_{\beta}$ is a Jordan canonical form. Furthermore, if we set $\beta_i=\beta \cap E_{\lambda_i}$ , then $[T\vert _{E_{\lambda_i}}]_{\beta_i}$ , the matrix representation of $T\vert _{E_{\lambda}}$ , the restriction of to $E_{\lambda_i}$ , is a Jordan canonical form. In other words,
$\displaystyle [T]_{\beta}=\begin{pmatrix} J_{1} & O & \cdots & O\ O & J_{2} &... ...O\ \vdots & \vdots & \ddots & \vdots \ O & O & \cdots & J_{n} \end{pmatrix}$
where each $J_i=[T\vert _{E_{\lambda_i}}]_{\beta_i}$ is a Jordan canonical form, and is a zero matrix.
Conversely, for each $E_{\lambda_i}$ , there exists an ordered basis $\beta_i$ for $E_{\lambda_i}$ such that $J_i:=[T\vert _{E_{\lambda_i}}]_{\beta_i}$ 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 such that $[T]_{\beta}$ is a Jordan canonical form, being the direct sum of matrices .
Each above can be further decomposed into Jordan blocks, and it turns out that the number of Jordan blocks in each is the dimension of $W_{\lambda_i}$ , the eigenspace of corresponding to $\lambda_i$ .