Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

A Jordan eigenpair residual can be much smaller than its eigenvalue error

Example

Let δ=10−8, J=(0100), λ^=10−4, and v=(1,λ^)T/1+λ^2. Then ∥v∥2=1 and

∥Jv−λ^v∥2=λ^21+λ^2<10−8,

but dist⁡(λ^,σ(J))=10−4. The residual is four decimal orders smaller than the eigenvalue error; a small residual does not give a same-size forward eigenvalue bound for this nonnormal matrix.

Facts & Assumptions

Given: The displayed J,δ,λ^,v.

[F1]

Eigenvalues, eigenvectors, eigenspaces Eλ(T)=ker⁡(T−λI), and the spectrum σF(T) of an endomorphism: eigenvalues are the scalars with a nonzero eigenvector.

[F2]

Closed pseudospectrum equals the resolvent, singular-value, and perturbation tests: a unit-vector residual at most ϵ places its scalar in the closed ϵ-pseudospectrum.

Verification

1.1givenalgebra

Direct multiplication gives Jv=(λ^,0)T/1+λ^2 and λ^v=(λ^,λ^2)T/1+λ^2. Their difference has norm λ^2/1+λ^2<10−8, and ∥v∥2=1.

2.1givenF2step 1.1algebra

Set r=(J−λ^I)v and E=−rv∗. For every unit vector x, ∥Ex∥2=∥r∥2∣v∗x∣≤∥r∥2, with equality at x=v, so ∥E∥2=∥r∥2<δ. Also (J+E)v=λ^v. By [F2], λ^∈Λδ(J).

3.1givenF1algebra∎

If λ≠0 and (J−λI)(x1,x2)T=0, the second row gives −λx2=0, hence x2=0; the first row then gives −λx1=0, hence x1=0. But e1 is a nonzero eigenvector for λ=0. Thus [F1] gives σ(J)={0} and dist⁡(λ^,σ(J))=10−4, which is 104 times the residual scale.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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