Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

A residual threshold on a normalised approximate eigenpair is exactly a normwise backward-error stopping rule

Statement

Let x be a unit vector and let r=Ax−μx. Then for every tolerance ε≥0,

∥r∥2≤ε⟺η(A,μ,x)≤ε.

So a residual stopping rule is exactly a normwise backward-error stopping rule.

Facts & Assumptions

Given: A unit vector x, a scalar μ, a matrix A, and the residual r=Ax−μx.

[L1]

The residual and backward error are the quantities of The residual r=Ax−μx and the normwise backward error of an approximate eigenpair.

[L2]

For unit x, the backward error equals the residual norm: η(A,μ,x)=∥Ax−μx∥2 (For a unit vector x, the smallest perturbation making (μ,x) an exact eigenpair has spectral norm ∥Ax−μx∥2).

Proof

technique · direct
1.1

If ∥r∥2≤ε, then [L2] gives η(A,μ,x)=∥r∥2≤ε.

L2algebra
1.2

If η(A,μ,x)≤ε, then [L2] again gives ∥r∥2=η(A,μ,x)≤ε.

L2algebra
2.1

Steps 1.1 and 1.2 prove the equivalence, and [L1] identifies it as a backward-error stopping rule.

L1step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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