Editing Bilinear form (section)

==Coordinate representation==
Let {{math|''V''}} be an {{mvar|n}}-[[dimension (vector space)|dimensional]] vector space with [[basis (linear algebra)|basis]] {{math|{'''e'''<sub>1</sub>, …, '''e'''<sub>''n''</sub>}<nowiki/>}}.

The {{math|''n'' × ''n''}} matrix ''A'', defined by {{math|1=''A<sub>ij</sub>'' = ''B''('''e'''<sub>''i''</sub>, '''e'''<sub>''j''</sub>)}} is called the ''matrix of the bilinear form'' on the basis {{math|{'''e'''<sub>1</sub>, …, '''e'''<sub>''n''</sub>}<nowiki/>}}.

If the {{math|''n'' × 1}} matrix {{math|''x''}} represents a vector {{math|'''x'''}} with respect to this basis, and similarly, the {{math|''n'' × 1}} matrix {{math|''y''}} represents another vector {{math|'''y'''}}, then:
<math display="block">B(\mathbf{x}, \mathbf{y}) = \mathbf{x}^\textsf{T} A\mathbf{y} = \sum_{i,j=1}^n x_i A_{ij} y_j. </math>

A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are all [[congruent matrices|congruent]]. More precisely, if {{math|{'''f'''<sub>1</sub>, …, '''f'''<sub>''n''</sub>}<nowiki/>}} is another basis of {{mvar|V}}, then
<math display="block">\mathbf{f}_j=\sum_{i=1}^n S_{i,j}\mathbf{e}_i,</math>
where the <math>S_{i,j}</math> form an [[invertible matrix]] {{mvar|S}}. Then, the matrix of the bilinear form on the new basis is {{math|''S''<sup>T</sup>''AS''}}.