Unimodular Form - Symmetric, Skew-symmetric and Alternating Forms

Symmetric, Skew-symmetric and Alternating Forms

We define a form to be

• symmetric if B(v, w) = B(w, v) for all v, w in V;
• alternating if B(v, v) = 0 for all v in V;
• skew-symmetric if B(v, w) = −B(w, v) for all v, w in V;

Proposition: Every alternating form is skew-symmetric. Proof: this can be seen by expanding B(v+w, v+w).

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 as a symmetric form and there exist symmetric/skew-symmetric forms which are not alternating.

A bilinear form is symmetric (resp. skew-symmetric) if and only if its coordinate matrix (relative to any basis) is symmetric (resp. skew-symmetric). A bilinear form is alternating if and only if 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 if and only if the maps B₁, B₂: V → V* are equal, and skew-symmetric if and only if they are negatives of one another. If char(F) ≠ 2 then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows

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

Also if char(F) ≠ 2 then one can define a quadratic form in terms of its associated symmetric form. One can likewise define quadratic forms corresponding to skew-symmetric forms, Hermitian forms, and skew-Hermitian forms; the general concept is ε-quadratic form.