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# 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. $W_{{\lambda}}\subseteq E_{{\lambda}}$, 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. 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. 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. 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 $v_{i}<v_{j}$ for $v_{i}\in\beta_{i}$ and $v_{j}\in\beta_{j}$ for $i<j$, 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. Prentice-Hall Inc., 1997.

## Mathematics Subject Classification

15A18*no label found*

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