Definition:Symmetric Bilinear Form

Definition

Let $R$ be a ring

Let $M$ be an $R$-module.

Let $b: M \times M \to R$ be a bilinear form.

Then $b$ is symmetric if and only if:

$\forall v, w, \in M: \map b {v, w} = \map b {w, v}$

Nondegenerate Symmetric Bilinear Form

Let $\Bbb K$ be a field.

Let $V$ be a vector space over $\Bbb K$.

Let $b: V \times V \to \Bbb K$ be a symmetric bilinear form.

Let $b$ be a nondegenerate bilinear form.

Then $b$ is a nondegenerate symmetric bilinear form.

Also see

• Definition:Quadratic Form
• Definition:Alternating Bilinear Form
• Definition:Reflexive Bilinear Form

Sources

• Weisstein, Eric W. "Symmetric Bilinear Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SymmetricBilinearForm.html