Alphabeta MathHow 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.
The complex induced infinity matrix norm
Definition
For with , define
For a vector , define . The absolute value is the complex modulus . The subscript on the matrix norm emphasizes that entries and vectors are complex; this definition is not an appeal to the real-only induced matrix p-norm convention. For square matrices the identity has norm .
Depends on
Used by
- Finite complex contour resolvent validator with cover and inconclusive statuses Algorithm
- Finite complex-contour validation of Floquet multiplier clusters Algorithm
- Finite-order Padé scaling-and-squaring exponential Algorithm
- Interval Krawczyk linear system certifier with explicit statuses Algorithm
- A complex infinity-norm defect bounds the inverse norm Lemma
- The complex infinity matrix norm is compatible and submultiplicative Lemma
- A resolvent contour bound transfers scalar rational error to matrices Theorem
- Diskwise Neumann certificates bound a complex matrix resolvent Theorem
- Krawczyk inclusion gives solutions and contraction gives uniqueness Theorem
- Padé error and squaring defects give a conditional exponential error bound Theorem
Dependency tree · two levels
3 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), §§4.2 and 10.5 (standard reference, not scraped)
- Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §5.1 (standard reference, not scraped)