---
tag: 0P24YW565F
kind: definition
name: Matrix of a bilinear form
msc: [15A63]
defines: 0QK6K857HS
uses: [0P799D6AMN, 0VTQTPQGAV]
sources:
  - url: https://en.wikipedia.org/wiki/Bilinear_form
    locator: Coordinate representation
---

## Definition

Let $F$ be a field and let $V$ be a vector space over $F$ with a [[basis|0KCQ9Y3QKA]] $(e_1, \dots, e_n)$.
Let $B$ be a [[bilinear form|0SQZMXB6GP]] on $V$.
The *matrix* of $B$ in this basis is the $n \times n$ matrix $M$ with entries
$$M_{ij} = B(e_i, e_j) \qquad (1 \le i, j \le n).$$

## Remarks

If $x$ and $y$ are the coordinate vectors of $u$ and $v$, then $B(u, v) = x^T M y$.
The map $B \mapsto M$ is a linear bijection from the bilinear forms on $V$ to the $n \times n$ matrices over $F$.
For the dot product on $\R^n$ and the standard basis, $M$ is the identity matrix.

## Explanations

### L1 human

A bilinear form is fixed by its values on pairs of basis vectors.
We store these $n^2$ values in a matrix.
Then the form becomes a product of a row vector, the matrix and a column vector.