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.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The B-inner product, B-norm, and dual residual norm
Definition
Let , , be Hermitian positive definite, with Cholesky factor and lower triangular with positive diagonal. This factor exists and is unique by A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique. Define the linear-first B-inner product by the B-norm by and the B-dual residual norm by The B-orthogonality condition is ; since it is equivalent to .
Depends on
- Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal
- A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique
- Real and complex inner product spaces, with the inner product linear in the first argument
- The norm $\lVert v\rVert=\sqrt{\langle v,v\rangle}$ induced by a real or complex inner product
Used by
- Finite-budget block LOBPCG for a Hermitian-definite pencil with explicit rank-loss status Algorithm
- A principal-angle bound encloses the largest selected generalized Ritz value across a spectral gap Lemma
- Generalized Ritz values obey min-max bounds and are monotone under trial-space inclusion Lemma
- A B-dual residual bounds distance to the spectrum of a Hermitian-definite pencil Theorem
- A Hermitian-definite pencil has a real spectrum and a B-orthonormal eigenbasis Theorem
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
- Peter Arbenz, Lecture Notes on Solving Large Scale Eigenvalue Problems, §11.4 (standard reference, not scraped)