TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

Diskwise Neumann certificates bound a complex matrix resolvent

Statement

Let $n \geq 1$ , let $Γ$ be a nonempty finite union of closed line segments in $C$ , let $A \subseteq C^{n \times n}$ be nonempty, and let $A_{0} \in C^{n \times n}$ . Each segment of $Γ$ is assigned to one of finitely many disks $D_{j} = {z : ∣ z - z_{j} ∣ \leq r_{j}}$ whose radius is $r_{j} \geq 0$ and whose disk contains both endpoints of that segment. For each disk there are $R_{j} \in C^{n \times n}$ and nonnegative bounds $e_{j}, δ_{A}, s_{j}$ satisfying $e_{j} \geq ∥ I - R_{j} (z_{j} I - A_{0}) ∥_{\infty, C}, δ_{A} \geq \sup_{A \in A} ∥ A - A_{0} ∥_{\infty, C}, s_{j} \geq ∥ R_{j} ∥_{\infty, C} .$ If $q_{j} = e_{j} + s_{j} (r_{j} + δ_{A}) < 1$ for each $j$ , then every $z \in Γ$ and $A \in A$ make $z I - A$ invertible, and $∥ (z I - A)^{- 1} ∥_{\infty, C} \leq \max_{j} \frac{s_{j}}{1 - q_{j}} .$ The theorem bounds the resolvent on the supplied contour cover; it does not count enclosed eigenvalues or certify a nonnormal eigenvalue disk from a residual.

Facts & Assumptions

Given: $Γ, A, A_{0}, D_{j}, R_{j}, e_{j}, δ_{A}, s_{j}$ and $q_{j}$ as in the statement.

[F1]

A defect of complex infinity norm at most $q < 1$ has inverse norm at most $(1 - q)^{- 1}$ (A complex infinity-norm defect bounds the inverse norm).

[F2]

The complex infinity matrix norm is compatible with products (The complex infinity matrix norm is compatible and submultiplicative).

[F3]

A product of square matrices is nonsingular only if each factor is nonsingular, by rank-nullity (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

[F4]

The complex infinity matrix norm is the maximum absolute row sum, so $∥ I ∥_{\infty, C} = 1$ (The complex induced infinity matrix norm).

Proof

1.1givenalgebra

Each segment of $Γ$ has both endpoints in its assigned disk. A disk is convex, so the segment between its endpoints lies in the disk; the finite assignments therefore cover all of $Γ$ .

1.2givenF1F2F4algebra

Fix a disk, $z \in D_{j}$ , and $A \in A$ . The exact identity $I - R_{j} (z I - A) = I - R_{j} (z_{j} I - A_{0}) - R_{j} ((z - z_{j}) I - (A - A_{0}))$ and [F2] give $∥ I - R_{j} (z I - A) ∥_{\infty, C} \leq e_{j} + s_{j} (∣ z - z_{j} ∣ + ∥ A - A_{0} ∥_{\infty, C}) \leq q_{j} < 1$ . [F1] applies to $M = R_{j} (z I - A) = I - E$ with $E = I - M$ , giving $∥ M^{- 1} ∥_{\infty, C} \leq (1 - q_{j})^{- 1}$ . Here [F4] gives $∥ (z - z_{j}) I ∥_{\infty, C} = ∣ z - z_{j} ∣$ , and the row-sum norm's triangle inequality and compatibility [F2, F4] justify the defect estimate.

2.1givenF1F2F3F4step 1.1step 1.2algebra∎

By [F4], $∥ I ∥_{\infty, C} = 1$ . Since $M = R_{j} (z I - A)$ is nonsingular, both square factors are nonsingular by [F3]. Then $(z I - A)^{- 1} = M^{- 1} R_{j}$ , so [F2] gives $∥ (z I - A)^{- 1} ∥_{\infty, C} \leq s_{j} / (1 - q_{j})$ . The disk cover from step 1.1 yields the maximum bound on all of $Γ$ , uniformly in $A$ .

Depends on

Used by

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

Siegfried M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica 19 (2010), §§10.5–10.6 (standard reference, not scraped)
Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §5.1 (standard reference, not scraped)