Bilinear form

From Maths
Jump to: navigation, search

This is a specialisation of a Bilinear map and a generalisation of Inner product


Let [ilmath](V,F)[/ilmath] be a vector space over a field [ilmath]F[/ilmath], a mapping:

  • [math]\langle\cdot,\cdot\rangle:V\times V\rightarrow F[/math]

is a bilinear form[1] if:

  • It is bilinear, that is to say:
    • It is linear in each coordinate, which is to say:
      • [math]\forall x,y,z\in V\ \forall\alpha,\beta\in F[\langle\alpha x+\beta y,z\rangle=\alpha\langle x,z\rangle+\beta\langle y,z\rangle][/math] and
      • [math]\forall x,y,z\in V\ \forall\alpha,\beta\in F[\langle x,\alpha y+\beta z\rangle=\alpha\langle x,y\rangle+\beta\langle x,z\rangle][/math]


We say the bilinear form is property when it has any of the following properties:

Property Definition Comment
Symmetric[1] [math]\forall x,y\in V[\langle x,y\rangle=\langle y,x\rangle][/math]
Skew-symmetric[1] or Antisymmetric [math]\forall x,y\in V[\langle x,y\rangle=-\langle y,x\rangle][/math] I use antisymmetric
Alternate[1] or Alternating [math]\forall x\in V[\langle x,x\rangle=0][/math] I use alternating. [ilmath]0[/ilmath] denotes the additive identity of [ilmath]F[/ilmath]

See also

TODO: Unite with inner product over real or complex field. Page 260 in Roman and page 206 are what is needed


  1. 1.0 1.1 1.2 1.3 Advanced Linear Algebra - Steven Roman - Third Edition - Springer - Graduate Texts in Mathematics