DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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.

Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism

Definition

Let $V$ be a vector space over a field $F$ and let $T : V \to V$ be linear. A scalar $λ \in F$ is an eigenvalue of $T$ when there is a nonzero vector $v \in V$ such that

$T (v) = λ v .$

Such a vector is an eigenvector belonging to $λ$ . The eigenspace belonging to $λ$ is

$E_{λ} (T) : = \ker (T - λ I_{V}) = {v \in V : T (v) = λ v} .$

Thus $E_{λ} (T)$ is a linear subspace and always contains $0_{V}$ , while $λ$ is an eigenvalue exactly when $E_{λ} (T) \neq {0_{V}}$ . The spectrum of $T$ over $F$ is

$σ_{F} (T) : = {λ \in F : λ is an eigenvalue of T} .$

In particular, the unique endomorphism of the zero space has empty spectrum.

Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism

Definition

Depends on

Dependency tree · 2 levels

Sources

Eigenvalues, eigenvectors, eigenspaces Eλ(T)=ker⁡(T−λI)Eλ​(T)=ker(T−λI), and the spectrum σF(T)σF​(T) of an endomorphism

Definition

Depends on

Dependency tree · 2 levels

Sources

Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism