DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

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.

Sesquilinear and Hermitian forms over a field with an involution, using the convention linear in the first variable

Definition

Let $F$ be a field with an involution, an automorphism $σ : F \to F$ satisfying $σ^{2} = {id}_{F}$ . A function $H : V \times V \to F$ is sesquilinear, with the convention used here, when it is linear in the first variable and $σ$ -linear in the second:

$H (a u + b v, w) = a H (u, w) + b H (v, w),$

$H (u, a v + b w) = σ (a) H (u, v) + σ (b) H (u, w) .$

It is Hermitian when

$H (u, v) = σ (H (v, u))$

for all $u, v \in V$ . When $σ$ is the identity, sesquilinear forms are bilinear and Hermitian forms are symmetric bilinear forms.

Depends on

Used by

For the linear-first convention, a basis change by P sends a sesquilinear matrix A to P^mathsf TA σ(P); Hermitian forms satisfy A=σ(A)^mathsf T Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)