Editing Bilinear form (section)

===Symmetric, skew-symmetric, and alternating forms===
We define a bilinear form to be
* '''[[Symmetric bilinear form|symmetric]]''' if {{math|1=''B''('''v''', '''w''') = ''B''('''w''', '''v''')}} for all {{math|'''v'''}}, {{math|'''w'''}} in {{mvar|V}};
* '''[[Alternating form|alternating]]''' if {{math|1= ''B''('''v''', '''v''') = 0}} for all {{math|'''v'''}} in {{mvar|V}};
* '''{{visible anchor|skew-symmetric bilinear form|text=skew-symmetric}}''' or '''{{visible anchor|antisymmetric bilinear form|text=antisymmetric}}''' if {{math|1=''B''('''v''', '''w''') = −''B''('''w''', '''v''')}} for all {{math|'''v'''}}, {{math|'''w'''}} in {{mvar|V}};
*; Proposition: Every alternating form is skew-symmetric.
*; Proof: This can be seen by expanding {{math|''B''('''v''' + '''w''', '''v''' + '''w''')}}.

If the [[Characteristic (algebra)|characteristic]] of {{mvar|K}} is not 2 then the converse is also true: every skew-symmetric form is alternating. However, if {{math|1=char(''K'') = 2}} then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetric forms that are not alternating.

A bilinear form is symmetric (respectively skew-symmetric) [[if and only if]] its coordinate matrix (relative to any basis) is [[Symmetric matrix|symmetric]] (respectively [[Skew-symmetric matrix|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 {{math|char(''K'') ≠ 2}}).

A bilinear form is symmetric if and only if the maps {{math|''B''<sub>1</sub>, ''B''<sub>2</sub>: ''V'' → ''V''<sup>∗</sup>}} are equal, and skew-symmetric if and only if they are negatives of one another. If {{math|char(''K'') ≠ 2}} then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows
<math display="block">B^{+} = \tfrac{1}{2} (B + {}^{\text{t}}B)  \qquad  B^{-} = \tfrac{1}{2} (B - {}^{\text{t}}B) ,</math>
where {{math|<sup>t</sup>''B''}} is the transpose of {{math|''B''}} (defined above).