An alternating bilinear form on a vector space 
over a field 
is a bilinear form 
satisfying 
for every 
. It follows that 
. When the characteristic
of 
is not 2, this condition is equivalent to skew-symmetry. A nondegenerate alternating
bilinear form is a symplectic
form on the vector space. If the vector
space is finite-dimensional, its dimension is necessarily
even.
Alternating Bilinear Form
See also
Bilinear Form, Symplectic Form, Vector SpaceExplore with Wolfram|Alpha
References
Bourbaki, N. Algebra I: Chapters 1-3. Berlin, Germany: Springer-Verlag, 1989.Cite this as:
Weisstein, Eric W. "Alternating Bilinear Form." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlternatingBilinearForm.html