Generalised eigenspace

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 $\lambda \in F$. The generalised eigenspace of $T$ for $\lambda$ is $$V_\lambda = \{v \in V : (T - \lambda \id_V)^k (v) = 0 \text{ for some } k \ge 1\}.$$

Remarks

It is a $T$-invariant subspace that contains the eigenspace $E_\lambda$. If $\dim V = n$ is finite, then $V_\lambda = \ker (T - \lambda \id_V)^n$. Its dimension is the algebraic multiplicity of $\lambda$, in finite dimensions. Its nonzero elements are the generalised eigenvectors for $\lambda$.

Explanations

L1 human

The generalised eigenspace is a larger version of the eigenspace. It contains the vectors that become zero after we apply $T - \lambda \id$ several times. These spaces always fill the space over $\C$, even when the eigenspaces do not.