DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Ritz values and Ritz vectors extracted from the Arnoldi Hessenberg reduction

Definition

Let Arnoldi run through step $m$ and write

$H_{m} : = Q_{m}^{*} A Q_{m} .$

An eigenvalue $θ$ of $H_{m}$ is a Ritz value of $A$ from the Arnoldi subspace $K_{m} (A, b)$ .

If $y \neq 0$ satisfies

$H_{m} y = θ y,$

then the lifted vector

$u : = Q_{m} y$

is the corresponding Ritz vector.

Depends on

Used by

An Arnoldi Ritz pair has residual norm controlled by the last Hessenberg subdiagonal entry Proposition

Dependency tree · two levels

8 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

Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra (standard reference, not scraped)