Correspondence between quadratic forms and alternating bilinear forms at 2

Let $𝕂$ be a field with [[Characteristic| $c h a r 𝕂 = 2$ ]] and $𝑉$ be a vector space over $𝕂$ .

For every quadratic form $𝑞 : 𝑉 \to 𝕂$ the polar form

𝑏 𝑞 : 𝑉 \times 𝑉 \to 𝕂 (𝑣, 𝑤) \mapsto 𝑞 (𝑣 + 𝑤) - 𝑞 (𝑣) - 𝑞 (𝑤)

is an alternating bilinear form.¹ ^P1 2. For every quadratic form $𝑞 : 𝑉 \to 𝕂$ there exists a (in general not unique) bilinear form $𝜀 0 : 𝑉 \times 𝑉 \to 𝕂$ such that $𝑞 (𝑣) = 𝜀 0 (𝑣, 𝑣)$ , and we have

𝑏 𝑞 (𝑣, 𝑤) = 𝜀 0 (𝑣, 𝑤) - 𝜀 0 (𝑤, 𝑣)

^P2 3. For every alternating bilinear form $𝑏 : 𝑉 \times 𝑉 \to 𝕂$ there exists a quadratic form $𝑞 : 𝑉 \to 𝕂$ such that $𝑏 𝑞 = 𝑏$ . The complete set of such quadratic forms is ${𝑞 + 𝜂 : 𝜂 \in 𝑉 *}$ .

Proof

^P1 follows immediately. Let $𝑏 : 𝑉 \times 𝑉 \to 𝕂$ be an alternating bilinear form with Gram matrix $𝐵$ , so that
$𝑏 (𝑣, 𝑤) = 𝑣 𝖳 𝐵 𝑤$
Noting that the diagonal entries of $𝐵$ must be zero, there exist unique strict upper and strict lower triangular matrices $𝐵 \pm$ respectively, so that
$𝐵 = 𝐵 + + 𝐵 -$
where $𝐵 - = 𝐵 𝖳 +$ . Then
$𝜀 0 (𝑣, 𝑤) = 𝑣 𝖳 𝐵 + 𝑣$
is a bilinear form and
$𝑞 (𝑣) = 𝜀 0 (𝑣, 𝑣) + 𝜂 (𝑣) = 𝑣 𝖳 𝐵 + 𝑣 + 𝜂 𝑣$
defines a quadratic form. Then
$𝑏 𝑞 (𝑣, 𝑤) = 𝑞 (𝑣 + 𝑤) - 𝑞 (𝑣) - 𝑞 (𝑤) + 𝜂 (𝑣 + 𝑤) - 𝜂 𝑣 - 𝜂 𝑤 = (𝑣 + 𝑤) 𝖳 𝐵 + (𝑣 + 𝑤) - 𝑣 𝖳 𝐵 + 𝑣 - 𝑤 𝖳 𝐵 + 𝑤 = 𝑣 𝖳 𝐵 + (𝑣 + 𝑤) + 𝑤 𝖳 𝐵 + (𝑣 + 𝑤) - 𝑣 𝖳 𝐵 + 𝑣 - 𝑤 𝖳 𝐵 + 𝑤 = 𝑣 𝖳 𝐵 + 𝑤 + 𝑤 𝖳 𝐵 + 𝑣 = 𝑣 𝖳 𝐵 + 𝑤 + 𝑣 𝖳 𝐵 - 𝑤 = 𝑣 𝖳 𝐵 𝑤 𝖳 = 𝑏 (𝑣, 𝑤)$
as required.

tidy | en | SemBr

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

Correspondence between quadratic forms and alternating bilinear forms at 2

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Correspondence between quadratic forms and alternating bilinear forms at 2

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