DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-16

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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

If the minimal polynomial splits, V is the direct sum of the stabilised generalised eigenspaces Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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