Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-16
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Definition

Let T:V→V be an endomorphism. For an irreducible polynomial q∈F[x] and an integer e≥1, the subspace

Vq,e(T):=ker⁡(q(T)e)=ker⁡((qe)(T))

is the qe-primary component of T when qe is the corresponding factor of μT.

Here q(T) uses polynomial evaluation at an endomorphism (Polynomial evaluation at an endomorphism: p(T)=∑kakTk).

For λ∈F, the generalised eigenspace of exponent e is

Gλ(e)(T):=ker⁡(T−λI)e.

Its first term is the ordinary eigenspace Gλ(1)(T)=Eλ(T) (Eigenvalues, eigenvectors, eigenspaces Eλ(T)=ker⁡(T−λI), and the spectrum σF(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

Dependency tree · 2 levels

Sources