DefinitionDefinition: Literature-sourcedProof: 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.

Primary matrix sign for spectra off the imaginary axis

Definition

Let $A \in C^{n \times n}$ , $n \geq 1$ , have $σ (A) \cap i R = \emptyset$ , where $σ (A)$ is the spectrum from Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism. The scalar function $s (z) = 1$ for $Re z > 0$ and $s (z) = - 1$ for $Re z < 0$ is locally constant and holomorphic on $C ∖ i R$ . Define the primary matrix sign by $sign (A) : = s (A)$ using Analytic primary matrix functions by Jordan blocks. This definition fixes the same half-plane branch on every Jordan block belonging to an eigenvalue.

Depends on

Used by

Dependency tree · two levels

12 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

Jan Zur, Matrix Functions, TU Berlin full lecture notes (standard reference, not scraped)
Nicholas J. Higham, Functions of Matrices: Theory and Computation, Chapter 5: Matrix Sign Function (standard reference, not scraped)