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.

Closed epsilon-pseudospectrum in the spectral norm

Definition

For a complex matrix $A \in C^{n \times n}$ with $n \geq 1$ and $ϵ > 0$ , its closed $ϵ$ -pseudospectrum is

$Λ_{ϵ} (A) : = {z \in C : σ_{\min} (z I - A) \leq ϵ},$

where $σ_{\min}$ is the smallest singular value from The singular values of a linear map as the eigenvalues of the positive square root of T^*T. This is the unstructured complex perturbation convention in the spectral norm The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||. In particular, real $A$ is regarded as a complex matrix and admissible perturbations may be complex. The non-strict inequality fixes the boundary: points with $σ_{\min} (z I - A) = ϵ$ are included.

Depends on

Used by

Dependency tree · two levels

9 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

Youcef Saad, Numerical Methods for Large Eigenvalue Problems, 2nd ed., §3.5 (standard reference, not scraped)
Mark Embree, Matrix Theory course notes, §5.3 (standard reference, not scraped)