Bilinear mapping

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In the mathematical sub-area of linear algebra and related areas, the bilinear mappings generalize the most diverse concepts of products (in the sense of a multiplication ). The bilinearity corresponds to the distributive law

with normal multiplication.


A -bilinear mapping is a 2- multilinear mapping , that is, a mapping

, where , and are three modules or vector spaces above the (same) ring or body ,

such that for each (solid selected) from

a - is linear mapping , and for each off

is a linear map . For any , and therefore holds

One can say that the concept of bilinearity represents a generalization of the (left and right) distributive laws that apply to rings and especially to bodies . The bilinearity describes not only (like the distributive laws) the behavior of the mapping with regard to addition, but also with regard to scalar multiplication .

More precisely: If a (possibly non-commutative) ring is included , then the side of the modules must be taken into account, i.e. H. must be a right and a left module. The sidedness of remains freely selectable (in the equations it is on the left).

Continuity and differentiability

Bilinear mappings with a finite dimensional domain are always continuous .

If a bilinear mapping is continuous, it is also totally differentiable and it applies

Using the chain rule it follows that two differentiable functions, which are linked with a bilinear mapping , can be derived with the generalization of the product rule : If functions are totally differentiable, then it holds


All common products are bilinear mappings: the multiplication in a field (real, complex , rational numbers ) or a ring ( whole numbers , matrices), but also the vector or cross product , and the scalar product on a real vector space.

The bilinear forms are a special case of bilinear mapping . With these, the range of values ​​is identical to the scalar field of the vector spaces and .

Bilinear forms are important for analytic geometry and duality theory.

In image processing, bilinear filtering is used for interpolation.

Other properties

Symmetry and antisymmetry (for ) and other properties are defined as in the more general case of multilinear maps .

A bilinear map turns into an algebra .

In the case of complex vector spaces, one also considers sesquilinear (“one and a half” -linear) mappings, which are antilinear in the second (or possibly in the first) argument, i.e.

(where denotes the complex conjugation ) while all other equations above remain.

Relation to tensor products

Bilinear maps are classified by the tensor product in the following sense : Is

a bilinear mapping, there is a uniquely determined linear mapping

conversely, defines any linear map

a bilinear map

These two constructions define a bijection between the space of the bilinear map and the space of the linear map .

Bilinear mappings over finite-dimensional vector spaces

Are and finite -Vektorräume with randomly selected bases of , and by , then there is for any of the presentation

with coefficients from and analogously for any one from the representation

With the calculation rules of the bilinear mapping this then results

The bilinear mapping is thus determined by the images of all ordered pairs of the basis vectors of and . If there is also a K-vector space, the image spans a maximally dimensional sub-vector space of . In general, however, the image of a bilinear mapping between vector spaces is not a sub-vector space.

For bilinear forms the are off , so that they can be noted in a matrix in an obvious way. This matrix is ​​then the coordinate representation of the bilinear form with respect to the selected bases.