DefinitionDefinition: AI-adaptedProof: 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.

The basis-independent characteristic polynomial $χ_{T}$ of an endomorphism of a finite-dimensional space, including $χ_{T} = 1$ in dimension zero

Definition

Let $T : V \to V$ be an endomorphism of a finite-dimensional vector space over $F$ . Choose an ordered basis $B$ and define

$χ_{T} (x) : = χ_{[T]_{B}^{B}} (x) .$

Matrices representing the same endomorphism in different ordered bases are similar, and similar matrices have the same characteristic polynomial. Hence $χ_{T}$ is independent of $B$ . When $V = {0_{V}}$ , the empty ordered basis gives $χ_{T} = 1$ .

The basis-independent characteristic polynomial $χ_{T}$ of an endomorphism of a finite-dimensional space, including $χ_{T} = 1$ in dimension zero

Definition

Depends on

Dependency tree · 2 levels

Sources

The basis-independent characteristic polynomial χTχT​ of an endomorphism of a finite-dimensional space, including χT=1χT​=1 in dimension zero

Definition

Depends on

Dependency tree · 2 levels

Sources

The basis-independent characteristic polynomial $χ_{T}$ of an endomorphism of a finite-dimensional space, including $χ_{T} = 1$ in dimension zero