Bilinear form/Introduction/Section

Real inner products are positive definite symmetric bilinear forms. In the following, we discuss bilinear forms in general. Beside inner products, there are the Hesse-forms, which are important in higher dimensional analysis in order to determine extrema, and the Minkowski-forms, which are used to describe special relativity theory.

Definition

Let ${}K$ be a field, and let ${}V$ denote a ${}K$-vector space. A mapping

$V\times V\longrightarrow K,(v,w)\longmapsto \left\langle v,w\right\rangle ,$

is called a bilinear form, if, for all ${}v\in V$, the induced mappings

$V\longrightarrow K,w\longmapsto \left\langle v,w\right\rangle ,$

and for all ${}w\in V$, the induced mappings

$V\longrightarrow K,v\longmapsto \left\langle v,w\right\rangle ,$

are

${}K$-linear.

Bilinear simply means being multilinear in two components. An extreme example is the zero form, which assigns to every pair the value ${}0$. It is easy to describe many different bilinear forms on ${}K^{n}$.

Example

Let ${}V=K^{n}$, and let ${}a_{ij}\in K$ denote fixed numbers, for ${}1\leq i,j\leq n$. Then, the assignment

$({\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}},{\begin{pmatrix}y_{1}\\\vdots \\y_{n}\end{pmatrix}})\longmapsto \Psi (x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})=\sum _{i,j}a_{ij}x_{i}y_{j}$

is a bilinear form. In case

${}a_{ij}=0\,$

for all ${}i,j$, this is the zero form; in case

${}a_{ij}=\delta _{ij}={\begin{cases}1,{\text{ if }}i=j\,,\\0{\text{ else}}\,,\end{cases}}\,$

we have the standard inner product (where the expression makes sense for every field; the property of being positive definite does only make sense for an ordered field). In case ${}n=4$ and

${}\Psi (x_{1},\ldots ,x_{4},y_{1},\ldots ,y_{4})=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}\,,$

we talk about a Minkowski-form. For ${}n=2$ and

${}\Psi (x_{1},x_{2},y_{1},y_{2})=x_{1}y_{2}-x_{2}y_{1}\,,$

this is the determinant in the two-dimensional case.

An important property of a bilinear form (which inner products fulfill) is formulated in the next definition.

Definition

Let ${}K$ be a field, and let ${}V$ denote a ${}K$-vector space. A bilinear form

$V\times V\longrightarrow K,(v,w)\longmapsto \left\langle v,w\right\rangle ,$

is called nondegenerate, if for every ${}v\in V$, ${}v\neq 0$, the induced mapping

$V\longrightarrow K,w\longmapsto \left\langle v,w\right\rangle ,$

and, for every ${}w\in V$, ${}w\neq 0$, the induced mapping

$V\longrightarrow K,v\longmapsto \left\langle v,w\right\rangle ,$

is not the

zero mapping.

We will prove in fact, for a vector space endowed with a nondegenerate bilinear form, the existence of a natural bijective relation between vectors and linear forms. This holds in particular for inner products. In general, there is a strong relation between bilinear forms and linear mappings to the dual space.

Lemma

Let ${}K$ be a field, let ${}V$ denote a ${}K$-vector space with its dual space ${}{V}^{*}$. Let

$\Theta \colon V\longrightarrow {V}^{*}$

be a linear mapping. Then

${}\Psi (u,v)=(\Theta (u))(v)\,$

defines a bilinear form

on ${}V$.

Proof

Since ${}\Theta (u)\in {V}^{*}$, the evaluation at a vector ${}v\in V$ yields an element of the base field. The linearity in the second component rests on the fact that ${}\Theta (u)$ belongs to the dual space. The linearity in the first component rest on the linearity of ${}\Theta$.

$\Box$