Generalised eigenvector

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, let $T \colon V \to V$ be an endomorphism, and let $\lambda \in F$. A generalised eigenvector of $T$ for $\lambda$ is a vector $v \in V$ with $v \ne 0$ and $$(T - \lambda \id_V)^k (v) = 0$$ for some integer $k \ge 1$. The least such $k$ is the rank of the generalised eigenvector.

Remarks

The generalised eigenvectors of rank $1$ are the eigenvectors. If a generalised eigenvector for $\lambda$ exists, then $\lambda$ is an eigenvalue. For $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the vector $e_2$ is a generalised eigenvector for $1$ of rank $2$, and it is not an eigenvector.

Explanations

L1 human

Some matrices do not have enough eigenvectors to make a basis. Generalised eigenvectors fill the gap: they reach zero after several applications of $T - \lambda \id$ instead of one. They are the building blocks of the Jordan normal form.