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.
Sesquilinear and Hermitian forms over a field with an involution, using the convention linear in the first variable
Definition
Let be a field with an involution, an automorphism satisfying . A function is sesquilinear, with the convention used here, when it is linear in the first variable and -linear in the second:
It is Hermitian when
for all . When is the identity, sesquilinear forms are bilinear and Hermitian forms are symmetric bilinear forms.
Depends on
Used by
- Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal Definition
- Real and complex inner product spaces, with the inner product linear in the first argument Definition
- For the linear-first convention, a basis change by P sends a sesquilinear matrix A to P^TA σ(P); Hermitian forms satisfy A=σ(A)^T Theorem
Dependency tree · two levels
8 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
- H. Pinkham, Linear Algebra, §7.8 (standard reference, not scraped)
- K. Conrad, Bilinear Forms, §7 (standard reference, not scraped)