DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-02

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.

Positive definite, negative definite, semidefinite, and indefinite quadratic forms

Definition

For a real $m \times m$ matrix $H$ , write $q_{H} (h) : = ⟨ H h, h ⟩$ using The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . It is positive definite when $q_{H} (h) > 0$ for every $h \neq 0$ , negative definite when $q_{H} (h) < 0$ for every $h \neq 0$ , positive semidefinite when $q_{H} (h) \geq 0$ for every $h$ , negative semidefinite when $q_{H} (h) \leq 0$ for every $h$ , and indefinite when it takes both positive and negative values. For a twice differentiable scalar field, its Hessian matrix is the matrix of The Hessian matrix and critical points of a scalar field.

Depends on

Used by

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Analysis, Convexity, and Optimization (standard reference, not scraped)