DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

Non-negative and positive operators

Let $V$ be a finite-dimensional real or complex inner product space and let $T : V \to V$ be linear. The operator $T$ is non-negative when $T$ is self-adjoint in the sense of Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space and

$⟨ T v, v ⟩ \geq 0$

for every $v \in V$ .

The operator $T$ is positive when $T$ is self-adjoint and

$⟨ T v, v ⟩ > 0$

for every nonzero $v \in V$ .

Depends on