Correspondence between quadratic forms and symmetric bilinear forms away from 2

Away from 2 every Quadratic form $𝑄 (𝑥)$ is associated with a symmetric Bilinear form $𝐵 (𝑥, 𝑦)$ via polarization

𝐵 𝑄 (𝑥, 𝑦) = 12 (𝑄 (𝑥 + 𝑦) - 𝑄 (𝑥) - 𝑄 (𝑦))

and vice versa by $𝑄 (𝑥) = 𝐵 𝑄 (𝑥, 𝑥)$ . #m/thm/general

Proof

Let $𝑄 (𝑥) = 𝑥 𝖳 𝐴 𝑥$ be a quadratic form and choose $𝐴$ to be symmetric. It follows

𝐵 (𝑥, 𝑦) = 12 (𝑄 (𝑥 + 𝑦) - 𝑄 (𝑥) - 𝑄 (𝑦)) = 12 ((𝑥 𝖳 + 𝑦 𝖳) 𝐴 (𝑥 + 𝑦) - 𝑥 𝖳 𝐴 𝑥 - 𝑦 𝖳 𝐴 𝑦) = 12 (𝑥 𝖳 𝐴 𝑦 + 𝑦 𝖳 𝐴 𝑥) = 𝑥 𝖳 𝐴 𝑦

Similarly for any bilinear form $𝐵 (𝑥, 𝑦) = 𝑥 𝖳 𝐴 𝑦$ , one can derive a quadratic form from $𝑄 (𝑥) = 𝐵 (𝑥, 𝑥) = 𝑥 𝖳 𝐴 𝑥$ . Clearly these processes are inverses of each other.

#state/tidy | #lang/en | #SemBr