DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Residual and near-orthogonality data for an approximate Hermitian eigenbasis

Definition

Let $A$ be a nonempty family of Hermitian matrices in $C^{n \times n}$ , let $V \in C^{n \times n}$ , and let $Θ = diag (θ_{1}, \dots, θ_{n})$ with real $θ_{j}$ . A residual and orthogonality certificate is a tuple of nonnegative numbers $δ, ρ, M_{A}, M_{Θ}, τ$ such that, for every $A \in A$ , $∥ V^{*} V - I ∥_{2} \leq δ < 1, ∥ A V - V Θ ∥_{2} \leq ρ, ∥ A ∥_{2} \leq M_{A}, ∥ Θ ∥_{2} \leq M_{Θ}, 1 - \sqrt{1 - δ} \leq τ .$ The certificate uses exact inequalities or separately verified upper bounds. Its eigenvalue enclosure radius is $η = ρ + (M_{A} + M_{Θ}) τ$ . This definition does not assert that computed residuals or Gram matrices satisfy the bounds; that is a separate certification procedure.

Depends on

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||

Used by

Dependency tree · two levels

2 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

Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §§5.1–5.6 (standard reference, not scraped)