LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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.

Positive Sylvester separation bounds the inverse operator norm

Statement

For the Sylvester operator $S_{A, B}$ of Sylvester operator and Frobenius separation number, if $sep (A, - B) > 0$ , then every $X \in F^{m \times n}$ satisfies

∥ X ∥_{F} \leq \frac{∥ A X + X B ∥_{F}}{sep (A, - B)} .

Facts & Assumptions

Given: $F \in {R, C}$ , $A \in F^{m \times m}$ , $B \in F^{n \times n}$ , and $sep (A, - B) > 0$ .

[F1]

Sylvester operator and Frobenius separation number: $sep (A, - B) = \min_{∥ Y ∥_{F} = 1} ∥ A Y + Y B ∥_{F}$ .

Proof

1.1givenchoosealgebra

If $X = 0$ , the claimed inequality is equality. If $X \neq 0$ , set $Y = X / ∥ X ∥_{F}$ , so $∥ Y ∥_{F} = 1$ and $S_{A, B} (Y) = S_{A, B} (X) / ∥ X ∥_{F}$ by linearity.

2.1F1step 1.1algebra∎

By [F1], $sep (A, - B) \leq ∥ S_{A, B} (Y) ∥_{F} = ∥ S_{A, B} (X) ∥_{F} / ∥ X ∥_{F}$ . Positivity of the separation permits multiplication and division by it, giving the desired bound.

Depends on

Sylvester operator and Frobenius separation number

Used by

Sylvester separation converts a residual into a forward error bound Theorem

Dependency tree · two levels

4 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)