Bilinear form: Encyclopedia II - Bilinear form - Symmetry

Bilinear form - Symmetry

A bilinear form B : V × V → F is said to be:

• symmetric if B(v,w) = B(w,v) for all
• skew-symmetric if B(v,w) = − B(w,v) for all (this is called skew-symmetric by mathematicians and antisymmetric by physicists)
• alternating if B(v,v) = 0 for all

Every alternating form is skew-symmetric; this may be seen by expanding

If the characteristic of F is not 2 then the converse is also true (every skew-symmetric form is alternating). If, however, char(F) = 2 then a skew-symmetric form is the same thing as a symmetric form and not all of these are alternating.

A bilinear form is symmetric (resp. skew-symmetric) iff its coordinate matrix (relative to any basis) is symmetric (resp. skew-symmetric). A bilinear form is alternating iff its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when char(F) ≠ 2).

A bilinear form is symmetric iff the maps are equal, and skew-symmetric iff they are negatives of one another. If char(F) ≠ 2 then one can always decompose a bilinear form into a symmetric and an skew-symmetric part as follows

where B* is the transpose of B (defined above).

Adapted from the Wikipedia article "Symmetry", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki