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. ${\displaystyle V,W}$ ${\displaystyle K}$ ${\displaystyle B\colon V\times W\to K}$

definition

There are vector spaces over a body (or more generally a left module and a right module over a not necessarily commutative ring ). ${\displaystyle V,W}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle W}$

An illustration

${\displaystyle B\colon V\times W\to K,\quad (v,w)\mapsto B(v,w)=\langle v,w\rangle }$

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

• ${\displaystyle \langle v_{1}+v_{2},w\rangle =\langle v_{1},w\rangle +\langle v_{2},w\rangle }$,
• ${\displaystyle \langle v,w_{1}+w_{2}\rangle =\langle v,w_{1}\rangle +\langle v,w_{2}\rangle }$,
• ${\displaystyle \langle \lambda v,w\rangle =\lambda \langle v,w\rangle }$,
• ${\displaystyle \langle v,w\lambda \rangle =\langle v,w\rangle \lambda }$.

There are , and . ${\displaystyle v,v_{1},v_{2}\in V}$ ${\displaystyle w,w_{1},w_{2}\in W}$ ${\displaystyle \lambda \in K}$

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: ${\displaystyle V}$ ${\displaystyle B(x,x),x\in V}$ ${\displaystyle x}$ ${\displaystyle B}$ ${\displaystyle B\colon V\times V\to K}$

• A bilinear form is called symmetric if ${\displaystyle B}$

${\displaystyle B(x,y)=B(y,x)}$

applies to all . ${\displaystyle x,y\in V}$

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 . ${\displaystyle 2\cdot B(x,y)=B(x+y,x+y)-B(x,x)-B(y,y)}$ ${\displaystyle K}$ ${\displaystyle 2}$ ${\displaystyle (\operatorname {char} (K)\neq 2)}$

• A bilinear form is called alternating if all form values ​​with respect to vanish, so if ${\displaystyle B}$ ${\displaystyle B}$

${\displaystyle B(x,x)=0}$

applies to all . ${\displaystyle x\in V}$

• A bilinear form is called antisymmetric or skew symmetric, if ${\displaystyle B}$

${\displaystyle B(x,y)=-B(y,x)}$

applies to all . ${\displaystyle x,y\in V}$

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 . ${\displaystyle \operatorname {char} (K)\neq 2}$ ${\displaystyle K=\mathbb {R} }$ ${\displaystyle K=\mathbb {C} }$

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 ${\displaystyle B}$ ${\displaystyle V}$ ${\displaystyle V}$

${\displaystyle V\times V\to \mathbb {R} ,\quad (x,y)\mapsto \mathrm {Re} \,B(x,y)}$

a symmetrical bilinear shape and

${\displaystyle V\times V\to \mathbb {R} ,\quad (x,y)\mapsto \mathrm {Im} \,B(x,y)}$

an alternating bilinear form.

• There is a canonical non-degenerate bilinear form

${\displaystyle V\times V^{*}\to K,\quad (v,f)\mapsto \langle v,f\rangle =f(v).}$

Degeneration space

Definition of the degeneration area

Be a bilinear form. The amount ${\displaystyle B\colon V\times W\to K}$

${\displaystyle ^{\perp }W\colon =\left\{v\mid \forall w\in W\colon B(v,w)=0\right\}\subseteq V}$

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 ${\displaystyle V}$ ${\displaystyle ^{\perp }W}$ ${\displaystyle W}$

${\displaystyle V^{\perp }\colon =\left\{w\mid \forall v\in V\colon B(v,w)=0\right\}\subseteq W}$

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 . ${\displaystyle B\colon V\times V\to K}$ ${\displaystyle B}$

The spellings and are also used for subsets or with an analogous definition . ${\displaystyle R^{\perp }}$ ${\displaystyle ^{\perp }S}$ ${\displaystyle R\subseteq V}$ ${\displaystyle S\subseteq W}$

Not degenerate bilinear shape

Each bilinear form defines two linear maps ${\displaystyle B}$

${\displaystyle B_{l}\colon V\to W^{*},\quad v\mapsto \left(w\mapsto B(v,w)\right)}$

and

${\displaystyle B_{r}\colon W\to V^{*},\quad w\mapsto \left(v\mapsto B(v,w)\right).}$

Right and left kernels are the kernels of these figures:

${\displaystyle \ker B_{l}={}^{\perp }W}$

${\displaystyle \ker B_{r}=V^{\perp }}$

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. ${\displaystyle B_{l}}$ ${\displaystyle B_{r}}$ ${\displaystyle B_{l}}$ ${\displaystyle B_{r}}$

The bilinear form has not degenerated if the following applies:

• For every vector there is a vector with and ${\displaystyle v\in V\setminus \{0\}}$ ${\displaystyle w\in W}$ ${\displaystyle B(v,w)\neq 0}$
• for each vector there is a vector with ${\displaystyle w\in W\setminus \{0\}}$ ${\displaystyle v\in V}$ ${\displaystyle B(v,w)\neq 0.}$

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 . ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle e=(e_{1},\ldots ,e_{n})}$ ${\displaystyle f=(f_{1},\ldots ,f_{m})}$

The representative matrix of a bilinear form is with ${\displaystyle B\colon V\times W\to K}$ ${\displaystyle M_{B}\in K^{n\times m}}$

${\displaystyle {(M_{B})}_{ij}:=B(e_{i},f_{j})}$.

If and are the coordinate vectors of and , then applies ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle v\in V}$ ${\displaystyle w\in W}$

${\displaystyle B(v,w)=x^{T}M_{B}\,y={\begin{pmatrix}x_{1}\dots x_{n}\end{pmatrix}}{\begin{pmatrix}B(e_{1},f_{1})&\cdots &B(e_{1},f_{m})\\\vdots &\ddots &\vdots \\B(e_{n},f_{1})&\dots &B(e_{n},f_{m})\end{pmatrix}}{\begin{pmatrix}y_{1}\\\vdots \\y_{m}\end{pmatrix}}}$,

where the matrix product provides a matrix, i.e. a body element. ${\displaystyle 1\times 1}$

Conversely, if any matrix is ​​defined, then ${\displaystyle M}$ ${\displaystyle n\times m}$

${\displaystyle B_{M}(x,y):=x^{T}M\,y}$

a bilinear form . ${\displaystyle B_{M}\colon K^{n}\times K^{m}\to K}$

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 ${\displaystyle e'}$ ${\displaystyle f'}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle {}_{e'}{\mathbf {1} }_{e}}$ ${\displaystyle e}$ ${\displaystyle e'}$ ${\displaystyle B}$

${\displaystyle A'={}_{e}{\mathbf {1} }_{e'}^{T}\cdot A\cdot {}_{f}{\mathbf {1} }_{f'}}$

If , and , then the matrices and are called congruent to one another . ${\displaystyle V=W}$ ${\displaystyle e=f}$ ${\displaystyle e'=f'}$ ${\displaystyle A}$ ${\displaystyle A'}$

Examples / properties

• The standard scalar product in has the unit matrix as its matrix with respect to the standard basis . ${\displaystyle \mathbb {R} ^{n}}$
• 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 . ${\displaystyle V=W}$ ${\displaystyle V}$ ${\displaystyle W}$
• 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. ${\displaystyle B\mapsto M_{B}}$ ${\displaystyle V\times W\to K}$ ${\displaystyle n\times m}$ ${\displaystyle K}$ ${\displaystyle (\lambda B_{1}+B_{2})(v,w):=\lambda B_{1}(v,w)+B_{2}(v,w)}$
• 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) ${\displaystyle \operatorname {char} (K)\neq 2}$
• 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 ) ${\displaystyle K=\mathbb {R} }$

Further remarks

• Bilinear forms correspond to linear maps ; see tensor product . ${\displaystyle V\times W\to K}$ ${\displaystyle V\otimes W\to K}$
• If the mapping does not necessarily take place in the scalar body , but in any vector space, one speaks of a bilinear mapping . ${\displaystyle K}$
• 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

Wikiversity: Course: Linear Algebra (Osnabrück 2015-2016) / Part II / Lecture 38  - Course materials

literature

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