Quadratic form

Correspondence between quadratic forms and alternating bilinear forms at 2

Let be a field with [[Characteristic|]] and be a vector space over .

1. For every quadratic form the polar form

is an alternating bilinear form.¹ ^P1 2. For every quadratic form there exists a (in general not unique) bilinear form such that , and we have

^P2 3. For every alternating bilinear form there exists a quadratic form such that . The complete set of such quadratic forms is .

Proof

^P1 follows immediately. Let be an alternating bilinear form with Gram matrix , so that

Noting that the diagonal entries of must be zero, there exist unique strict upper and strict lower triangular matrices respectively, so that

where . Then

is a bilinear form and

defines a quadratic form. Then

as required.

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Footnotes

1. Note that any minus signs in this Zettel could be replaced with plus signs. ↩