# Bilinear form

In linear algebra, a **bilinear form** is a function which assigns a scalar value to two vectors and which is linear in both of its arguments.

The two arguments can come from different vector spaces , but they must be based on a common scalar field ; a bilinear form is a map . A bilinear form is a linear form with regard to its first and its second argument, and thus in particular a multilinear form with two arguments.

## definition

There are vector spaces over a body (or more generally a left module and a right module over a not necessarily commutative ring ).

An illustration

is called bilinear form if the two conditions of a linear mapping (additivity and homogeneity) hold in both arguments:

- ,
- ,
- ,
- .

There are , and .

## Symmetry properties in the case V = W

If both arguments of the bilinear form come from the same vector space , it is called
the *form value of* the vector (with respect to ). The bilinear form can have additional symmetry properties:

- A bilinear form is called symmetric if

- applies to all .
- For a symmetrical bilinear form, ( polarization formula ). It follows that the bilinear is determined entirely by the entirety of the form of values, if the underlying field is a characteristic of unequal has .

- A bilinear form is called alternating if all form values with respect to vanish, so if

- applies to all .

- A bilinear form is called antisymmetric or skew symmetric, if

- applies to all .

Any alternating bilinear form is also antisymmetric. If , for example, what is fulfilled for and , then the reverse also applies: Every antisymmetric bilinear form is alternating. Looking more generally at modules over any commutative ring, these two terms are equivalent if the target module does not have a 2- torsion .

## Examples

- A scalar product on a real vector space is a non-degenerate, symmetrical, positively definite bilinear form.
- A scalar product on a complex vector space is not a bilinear form, but a sesquilinear form . However, if one understands as a real vector space, then

- a symmetrical bilinear shape and
- an alternating bilinear form.

- There is a canonical non-degenerate bilinear form

## Degeneration space

### Definition of the degeneration area

Be a bilinear form. The amount

is a subspace of and is called left kernel or left radical of the bilinear form. The symbolism " " should indicate that elements of the left core are precisely those which (in the sense of the bilinear form) are orthogonal to the entire space . Correspondingly means

Right-wing core or right-wing radical. If a bilinear form is symmetrical, the right kernel and left kernel match and this space is called the degeneration space of .

The spellings and are also used for subsets or with an analogous definition .

### Not degenerate bilinear shape

Each bilinear form defines two linear maps

and

Right and left kernels are the kernels of these figures:

If both cores trivial (the two images and therefore injective ), it means the bilinear **non-degenerate** or **non-degenerate** . Otherwise the bilinear form is called **degenerate** or **degenerate** . Are the pictures and even bijective , so isomorphisms , so is the bilinear **perfect pairing** . This always applies to finite-dimensional vector spaces, the terms **not degenerate** and **perfect** can therefore be used synonymously in this case.

The bilinear form has not degenerated if the following applies:

- For every vector there is a vector with and
- for each vector there is a vector with

If the bilinear form is symmetrical, then it has not degenerated if and only if its degeneration space is the zero vector space .

## Coordinate representation

For finite-dimensional and one can choose bases and .

The representative matrix of a bilinear form is with

- .

If and are the coordinate vectors of and , then applies

- ,

where the matrix product provides a matrix, i.e. a body element.

Conversely, if any matrix is defined, then

a bilinear form .

### Change of base

Are and further bases from and , continue the base change matrix from to . Then the matrix of in the new basis results as

If , and , then the matrices and are called congruent to one another .

### Examples / properties

- The standard scalar product in has the unit matrix as its matrix with respect to the standard basis .
- If and the same basis is used for and , then the following applies: The bilinear form is symmetric if and only if the matrix is symmetric , antisymmetric if and only if the matrix is antisymmetric , and alternating if and only if the matrix is alternating .
- The figure is a bijection of the space of the bilinear shapes onto the - matrices. If one defines the sum and scalar multiplication of bilinear forms in a canonical way ( ), this bijection is also a vector space isomorphism.
- For symmetrical bilinear forms over vector spaces of finite dimension there is a basis in which the representing matrix has a diagonal shape (if ). (see Gram-Schmidt's orthogonalization method for the special case of positively definite bilinear forms)
- If further , one can find a basis in which only the entries 1, −1 and 0 appear on the diagonal ( Sylvester's law of inertia )

## Further remarks

- Bilinear forms correspond to linear maps ;
*see*tensor product . - If the mapping does not necessarily take place in the scalar body , but in any vector space, one speaks of a bilinear mapping .
- The generalization of the concept of the bilinear form to more than two arguments is called the multilinear form .
- Over the field of complex numbers one often demands linearity in one argument and semi- linearity in the other; instead of a bilinear form, a sesquilinear form is obtained . In particular, an inner product over a real vector space is a bilinear form, but over a complex vector space it is only a sesquilinear form.

## Web links

## literature

- Gerd Fischer:
*Lineare Algebra*, Vieweg-Verlag, ISBN 3-528-03217-0