DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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.

The Gram matrix $G (v_{0}, \dots, v_{r - 1}) = (⟨ v_{i}, v_{j} ⟩)_{i, j < r}$ and Gram determinant, with empty value $1$

Definition

For a finite list $(v_{0}, \dots, v_{r - 1})$ in a real or complex inner product space (Real and complex inner product spaces, with the inner product linear in the first argument), its Gram matrix is

$G (v_{0}, \dots, v_{r - 1}) : = (⟨ v_{i}, v_{j} ⟩)_{i, j < r} \in M_{r} (F),$

using the square matrix space The vector space $M_{m \times n} (F) : = F^{m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations. Its Gram determinant is the determinant (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix) $\det G (v_{0}, \dots, v_{r - 1})$ . For the empty list, the Gram matrix is the unique $0 \times 0$ matrix and its determinant is $1$ .

Depends on

Used by

A Gram determinant is nonnegative and is positive exactly when the vector list is linearly independent Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 71 results over 15 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

Kenneth Hoffman and Ray Kunze, Linear Algebra, 2nd ed., p. 332, Theorem 7 (standard reference, not scraped)

The Gram matrix G(v0,…,vr−1)=(⟨vi,vj⟩)i,j<rG(v0​,…,vr−1​)=(⟨vi​,vj​⟩)i,j<r​ and Gram determinant, with empty value 11