nLab bilinear form

Definitions

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

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

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

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

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

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

• positive definite if $\langle x,x\rangle \gt 0$ if $x \neq 0$.

• negative definite if $\langle x,x\rangle \lt 0$ if $x \neq 0$.

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

Categorification and $n$POV

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

• 2-Hilbert space

• Calabi-Yau object, Calabi-Yau A-infinity category

Generalizations

From Ab to monoidal categories

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

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

$$f \circ ((\rho \circ (a \otimes v)) \otimes w) = \pi \circ (a \otimes (f \circ (v \otimes w))$$

and

$$f \circ (v \otimes (\rho \circ (a \otimes w))) = \pi \circ (a \otimes (f \circ (v \otimes w))$$

However, if $C$ 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 $v:I \to V$ and $w:I \to V$, we have $f \circ (v \otimes w) = f \circ (w \otimes v)$.

From Mod to monoidal categories

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

Let $(C, I, \otimes)$ be a monoidal category. Given an object $V$, every morphism $I \otimes V \to V$ is given by the composition of an endomorphism $V \to V$ with the left unitor $\lambda_V:I \otimes V \to V$. Then given an object $V \in C$, a bilinear form is just a morphism $f:V \otimes V \to I$. However, if $C$ 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 $(C, 1, \times)$, the binary forms $V \times V \to 1$ 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 $v:I \to V$ and $w:I \to V$, we have $f \circ (v \otimes w) = f \circ (w \otimes v)$.

Related concepts

• quadratic form, polarization identity

• sesquilinear form

• symplectic form, Kähler form, Hermitian form

• Hermitian K-theory

• characteristic element of a bilinear form

• hyperbolic functor

• Riemannian metric

• Pythagorean theorem

• Cauchy–Schwarz inequality

• Bochner's theorem

References

• John Milnor, Dale Husemöller, Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer (1973) [doi:10.1007/978-3-642-88330-9]

• Richard Elman, Nikita Karpenko, Alexander Merkurjev, Algebraic and Geometric Theory of Quadratic Forms, Colloquium Publication 56, AMS (2008) [ams:coll-56, pdf]

• Igor R. Shafarevich, Alexey O. Remizov: §6 in: Linear Algebra and Geometry (2012) [doi:10.1007/978-3-642-30994-6, MAA-review]

Lecture notes:

• Keith Conrad, Bilinear forms [pdf, pdf]