DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-16

How statement and proof provenance work

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Primary components $\ker q (T)^{e}$ and generalised eigenspaces $G_{λ}^{(e)} (T) = \ker (T - λ I)^{e}$

Definition

Let $T : V \to V$ be an endomorphism. For an irreducible polynomial $q \in F [x]$ and an integer $e \geq 1$ , the subspace

$V_{q, e} (T) : = \ker (q (T)^{e}) = \ker ((q^{e}) (T))$

is the $q^{e}$ -primary component of $T$ when $q^{e}$ is the corresponding factor of $μ_{T}$ .

Here $q (T)$ uses polynomial evaluation at an endomorphism (Polynomial evaluation at an endomorphism: $p (T) = \sum_{k} a_{k} T^{k}$ ).

For $λ \in F$ , the generalised eigenspace of exponent $e$ is

$G_{λ}^{(e)} (T) : = \ker (T - λ I)^{e} .$

Its first term is the ordinary eigenspace $G_{λ}^{(1)} (T) = E_{λ} (T)$ (Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism). These kernels are $T$ -invariant by For every polynomial $p$ , both $\ker p (T)$ and $im p (T)$ are $T$ -invariant.

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sheldon Axler, Linear Algebra Done Right, 4th ed., §§8A–8B (standard reference, not scraped)
Anthony W. Knapp, Basic Algebra, 2nd ed., Ch. V, §5 (standard reference, not scraped)

Primary components ker⁡q(T)ekerq(T)e and generalised eigenspaces Gλ(e)(T)=ker⁡(T−λI)eGλ(e)​(T)=ker(T−λI)e