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.

Sylvester operator and Frobenius separation number

Definition

For $F \in {R, C}$ , $A \in F^{m \times m}$ , $B \in F^{n \times n}$ with $m, n \geq 1$ , define the Sylvester operator $S_{A, B} : F^{m \times n} \to F^{m \times n}$ by

$S_{A, B} (X) = A X + X B .$

Equip $F^{m \times n}$ with the Frobenius norm of The Frobenius norm $∥ A ∥_{F} = (\sum_{i, j} ∣ a_{i j} ∣^{2})^{1 / 2}$ on real or complex matrices. The column-vectorization isometry of Column vectorization is an isometry and represents the Sylvester map by a Kronecker sum identifies this norm with the Euclidean norm on $F^{m n}$ . Define

$sep (A, - B) : = σ_{\min} (S_{A, B}) = \min_{∥ X ∥_{F} = 1} ∥ A X + X B ∥_{F} .$

Here $σ_{\min}$ denotes the smallest singular value of the finite-dimensional linear map under these inner products. The finite-dimensional singular value decomposition Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition ensures this minimum is well-defined. The argument $- B$ is part of the conventional label: the operator being measured is $X \mapsto A X + X B$ .

Depends on

Used by

Dependency tree · two levels

13 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

Valeria Simoncini, Computational Methods for Linear Matrix Equations, SIAM Review 58(3), 2016 (standard reference, not scraped)
Elias Jarlebring, Numerical Methods for Lyapunov Equations, lecture notes in numerical linear algebra, 2017 (standard reference, not scraped)