Bilinear form

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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.


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 .


  • 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


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.

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