DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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 $q_H(h):=\langle Hh,h\rangle$ using The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$ . It is positive definite when $q_H(h)>0$ for every $h\ne0$ , negative definite when $q_H(h)<0$ for every $h\ne0$ , positive semidefinite when $q_H(h)\ge0$ for every $h$ , negative semidefinite when $q_H(h)\le0$ 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

A definite quadratic form has a uniform signed bound on the Euclidean unit sphere Lemma

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 86 results over 19 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

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