nLab bilinear form

Bilinear forms

Context

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

Bilinear forms

Definitions

A bilinear form is simply a linear map ,:VVk\langle -,-\rangle\colon V \otimes V \to k out of a tensor product of kk-modules into the ring kk (typically taken to be a field).

It is called symmetric if x,y=y,x\langle x,y\rangle = \langle y,x\rangle for all x,yVx,y \in V. For variants on this, such as the property of being conjugate-symmetric, see inner product space.

It is called antisymmetric or skew-symmetric if x,y=y,x\langle x,y\rangle = -\langle y,x\rangle for all x,yVx,y \in V.

It is called alternating if x,x=0\langle x,x\rangle = 0 for all xVx \in V.

It is called nondegenerate if the mate VV *=hom(V,k)V \to V^\ast = \hom(V, k) is injective (a monomorphism).

Let k=k = \mathbb{R} be the real numbers. A symmetric bilinear form is called

Examples

Example

An inner product on a real vector space (not though generally on complex vector spaces, see Rem. ) is an example of a symmetric bilinear form. (For some authors, an inner product on a real vector space is precisely a positive definite symmetric bilinear form. Other authors relax the positive definiteness to nondegeneracy. Perhaps some authors even drop the nondegeneracy condition (citation?).)

Remark

In contrast to Exp. , beware that an inner product on a complex vector space is typically not taken to be a bilinear form, but a sesquilinear form (called a Hermitian form if positive definite).

Example

If f: nf \colon \mathbb{R}^n \to \mathbb{R} is a differentiable function of class C 2C^2, then the Hessian of ff at a point defines a symmetric bilinear form. It may be degenerate, but in Morse theory, a Morse function is a C 2C^2 function such that the Hessian at each critical point is nondegenerate.

Categorification and nnPOV

Concepts which relate to (non-degenerate) bilinear forms from the nPOV and/or categorifications of the concept of bilinear forms include

Generalizations

From Ab to monoidal categories

This concept could be generalized from the category of abelian groups to any monoidal category:

Let (C,I,)(C, I, \otimes) be a monoidal category, let (k,1,π)(k, 1, \pi) be a monoid object in CC, and let (V,ρ)(V, \rho) be a kk-module object in CC. kk itself is a kk-module object with the action being represented by the monoid binary operation π\pi. A bilinear form is simply an action-preserving morphism f:VVkf:V \otimes V \to k from the tensor product VVV \otimes V to kk: given morphisms a:Ika:I \to k, v:IVv:I \to V, and w:IVw:I \to V,

f((ρ(av))w)=π(a(f(vw))f \circ ((\rho \circ (a \otimes v)) \otimes w) = \pi \circ (a \otimes (f \circ (v \otimes w))

and

f(v(ρ(aw)))=π(a(f(vw))f \circ (v \otimes (\rho \circ (a \otimes w))) = \pi \circ (a \otimes (f \circ (v \otimes w))

However, if CC is not a Ab-enriched category, then the morphisms of the category are not linear and it would probably be better to call these morphisms binary forms or something similar.

Symmetric bilinear forms can similarly be generalized from the category of abelian groups to any monoidal category, where for all morphisms v:IVv:I \to V and w:IVw:I \to V, we have f(vw)=f(wv)f \circ (v \otimes w) = f \circ (w \otimes v).

From Mod to monoidal categories

This concept could be generalized from the category of modules to any monoidal category, since the ground ring kk is the tensor unit of the category of kk-modules:

Let (C,I,)(C, I, \otimes) be a monoidal category. Given an object VV, every morphism IVVI \otimes V \to V is given by the composition of an endomorphism VVV \to V with the left unitor λ V:IVV\lambda_V:I \otimes V \to V. Then given an object VCV \in C, a bilinear form is just a morphism f:VVIf:V \otimes V \to I. However, if CC is not a Ab-enriched category, then the morphisms of the category are not linear and it would probably be better to call these morphisms binary forms or something similar. In cartesian monoidal categories (C,1,×)(C, 1, \times), the binary forms V×V1V \times V \to 1 always exist and is unique, by the universal property of the terminal object.

Symmetric bilinear forms can similarly be generalized from the category of modules to any monoidal category, where for all morphisms v:IVv:I \to V and w:IVw:I \to V, we have f(vw)=f(wv)f \circ (v \otimes w) = f \circ (w \otimes v).

References

Lecture notes:

Last revised on November 27, 2023 at 21:31:31. See the history of this page for a list of all contributions to it.