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

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

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

Bibliography

1

Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.