---
tag: 0744CN4MDA
kind: definition
name: Symmetric bilinear form
msc: [15A63]
defines: 0AXZNE5XNH
uses: [0P799D6AMN]
sources:
  - url: https://en.wikipedia.org/wiki/Symmetric_bilinear_form
    locator: Formal definition
---

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$.
A [[bilinear form|0SQZMXB6GP]] $B$ on $V$ is *symmetric* when
$$B(u, v) = B(v, u) \qquad \text{for all } u, v \in V.$$

## Remarks

In each basis, $B$ is symmetric if and only if its matrix is a [[symmetric matrix|0VR671PHE5]].
The dot product on $\R^n$ is symmetric.
If the characteristic of $F$ is not $2$, symmetric bilinear forms correspond one to one with [[quadratic forms|0RBQS1JJCR]].

## Explanations

### L1 human

A symmetric bilinear form gives the same value for $(u, v)$ and for $(v, u)$.
Inner products on real spaces are the main examples.
Their matrices are symmetric.