nLab
bilinear form

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 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

  • positive definite if x,x>0\langle x,x\rangle \gt 0 if x0x \neq 0.

  • negative definite if x,x<0\langle x,x\rangle \lt 0 if x0x \neq 0.

Examples

  • A inner product on a real vector space 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?).)

  • If f: nf \colon \mathbb{R}^n \to \mathbb{R} is 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.

Revised on October 13, 2013 02:52:56 by Urs Schreiber (82.113.121.239)