Bilinear mapping

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

${\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}$

with normal multiplication.

definition

A -bilinear mapping is a 2- multilinear mapping , that is, a mapping ${\displaystyle K}$

${\displaystyle f\colon E\times F\to G}$, where , and are three modules or vector spaces above the (same) ring or body , ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$

such that for each (solid selected) from ${\displaystyle y}$ ${\displaystyle F}$

${\displaystyle x\mapsto f(x,y)}$

a - is linear mapping , and for each off ${\displaystyle K}$ ${\displaystyle E\to G}$ ${\displaystyle x}$ ${\displaystyle E}$

${\displaystyle y\mapsto f(x,y)}$

is a linear map . For any , and therefore holds ${\displaystyle F\to G}$ ${\displaystyle x,x'\in E}$ ${\displaystyle y,y'\in F}$ ${\displaystyle \alpha \in K}$

${\displaystyle {\begin{aligned}f(x+x',y)&=f(x,y)+f(x',y)\\f(x\cdot \alpha ,y)&=\alpha \cdot f(x,y)\\f(x,y+y')&=f(x,y)+f(x,y')\\f(x,\alpha \cdot y)&=\alpha \cdot f(x,y).\\\end{aligned}}}$

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). ${\displaystyle K}$ ${\displaystyle 1}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle K}$ ${\displaystyle G}$

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 ${\displaystyle B}$

${\displaystyle DB(x_{0},y_{0})(x,y)\;=\;B(x_{0},y)\,+\,B(x,y_{0})}$

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 ${\displaystyle f,g}$

${\displaystyle {\begin{aligned}DB(f(\cdot ),g(\cdot \cdot ))(x_{0},y_{0})(x,y)&=D(B\circ (f,g))(x_{0},y_{0})(x,y)\\&=B(Df(x_{0})x,g(y_{0}))+B(f(x_{0}),Dg(y_{0})y)\end{aligned}}}$

Examples

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 . ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle E}$ ${\displaystyle F}$

${\displaystyle f\colon E\times F\to K}$

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 . ${\displaystyle F=E}$

A bilinear map turns into an algebra . ${\displaystyle E\times E\to E}$ ${\displaystyle E}$

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.

${\displaystyle f(x,\alpha \cdot y)=\alpha ^{*}\cdot f(x,y)}$

(where denotes the complex conjugation ) while all other equations above remain. ${\displaystyle *}$

Relation to tensor products

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

${\displaystyle f\colon E\times F\to G}$

a bilinear mapping, there is a uniquely determined linear mapping

${\displaystyle E\otimes F\to G,x\otimes y\mapsto f(x,y);}$

conversely, defines any linear map

${\displaystyle \lambda \colon E\otimes F\to G}$

a bilinear map

${\displaystyle E\times F\to G,\quad (x,y)\mapsto \lambda (x\otimes y).}$

These two constructions define a bijection between the space of the bilinear map and the space of the linear map . ${\displaystyle E\times F\to G}$ ${\displaystyle E\otimes F\to G}$

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 ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle K}$ ${\displaystyle (b_{i})_{i=1,\dotsc ,n}}$ ${\displaystyle E}$ ${\displaystyle (c_{j})_{j=1,\dotsc ,m}}$ ${\displaystyle F}$ ${\displaystyle x}$ ${\displaystyle E}$

${\displaystyle x=\sum _{i}x_{i}b_{i}}$with coefficients from and analogously for any one from the representation ${\displaystyle x_{i}}$ ${\displaystyle K}$ ${\displaystyle y}$ ${\displaystyle F}$

${\displaystyle y=\sum _{j}y_{j}c_{j}.}$

With the calculation rules of the bilinear mapping this then results

${\displaystyle f(x,y)=\sum _{i}\sum _{j}x_{i}y_{j}f(b_{i},c_{j}).}$

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. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle \operatorname {Im} (f)}$ ${\displaystyle n\cdot m}$ ${\displaystyle G}$

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. ${\displaystyle f(b_{i},c_{j})}$ ${\displaystyle K}$

swell

• Gerd Fischer : Linear Algebra. 17th edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-0996-4 .