nLab
sesquilinear form

Contents

Context

Higher linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

On a complex vector space VV a sesquilinear map is a function of two arguments

,:V×V \langle -,-\rangle \;\colon\; V \times V \longrightarrow \mathbb{C}

which is a linear function in one argument (say the second) and complex anti-linear in the other.

Definition

A sesquilinear form ,\langle -,-\rangle is called

positive definite if v *,v>0 negative definite if v *,v<0 positive semi-definite if v *,v0 negative semi-definite if v *,v0}for allv0. \left. \array{ \text{positive definite} &if& \langle v^\ast,v \rangle \gt 0 \\ \text{negative definite} &if& \langle v^\ast,v \rangle \lt 0 \\ \text{positive semi-definite} &if& \langle v^\ast,v \rangle \geq 0 \\ \text{negative semi-definite} &if& \langle v^\ast,v \rangle \leq 0 } \right\rbrace \;\; \text{for all}\; v \neq 0 \,.

Finally, it is called indefinite if it is neither positive nor negative semi-definite.

More generally:

Let AA be a star algebra. Then every left AA-module ρ l:AVV\rho_l \colon A \otimes V \longrightarrow V canonically becomes a right module ρ r:VAA\rho_r \colon V \otimes A \longrightarrow A by setting

ρ r(v,a)a *v \rho_r(v,a) \coloneqq a^\ast v

and vice versa.

With this operation understood to turn a left module VV into a right module, then a sesquilinear form on VV is simply an element in the tensor product of modules

V * AV * V^\ast \otimes_A V^\ast

of the AA-linear dual V *V^\ast of VV with itself.

References

See also:

Last revised on May 3, 2021 at 23:54:38. See the history of this page for a list of all contributions to it.