Bilinear form - Citizendia

In mathematics, a bilinear form on a vector space V is a bilinear mapping V × V → F, where F is the field of scalars. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a bilinear map is a function of two arguments that is linear in each In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division That is, a bilinear form is a function B: V × V → F which is linear in each argument separately:

$\begin{array}{l} \text{1. }B(u + u',v) = B(u,v) + B(u',v)\text{,} \\[4pt] \text{2. }B(u,v + v') = B(u,v) + B(u,v')\text{,} \\[4pt] \text{3. }B(\lambda u,v) = B(u, \lambda v) = \lambda\,B(u,v)\text{.} \\[4pt] \end{array}$

Any bilinear form on Fⁿ can be expressed as

$B(\textbf{x},\textbf{y}) = \textbf{x}^{\mathrm{T}}A\textbf{y} = \sum_{i,j=1}^n a_{ij} x_i y_j$

where A is an n × n matrix. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

The definition of a bilinear form can easily be extended to include modules over a commutative ring, with linear maps replaced by module homomorphisms. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars When F is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a sesquilinear form on a Complex vector space V is a map V × V &rarr C that is linear In Mathematics, a mapping f: V → W from a Complex vector space to another is said to be antilinear (or conjugate-linear

1 Coordinate representation
2 Maps to the dual space
3 Reflexivity and orthogonality
4 Different spaces
5 Relation to tensor products
6 On normed vector spaces
7 See also
8 References
9 External links

Coordinate representation

Let $C=\{e_{1},\ldots,e_{n}\}$ be a basis for a finite-dimensional space V. Define the $n\times n$ - matrix A by $(A i j) = B (e i, e j)$ . Then if the $n\times 1$ matrix x represents a vector v with respect to this basis, and analogously, y represents w, then:

$B(v,w) = x^{T} A y\,$

Suppose C' is another basis for V, with : $\begin{bmatrix}e'_{1} & \cdots & e'_{n}\end{bmatrix} = \begin{bmatrix}e_{1} & \cdots & e_{n}\end{bmatrix}S$ with S an invertible $n\times n$ - matrix. Now the new matrix representation for the symmetric bilinear form is given by :

$A' = S T A S$

Maps to the dual space

Every bilinear form B on V defines a pair of linear maps from V to its dual space V*. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals Define $B_1,B_2\colon V \to V^*$ by

B 1 (v)(w) = B (v, w)

B 2 (v)(w) = B (w, v)

This is often denoted as

$B_1(v) = B(v,{\cdot})$

$B_2(v) = B({\cdot},v)$

where the ( $\cdot$ ) indicates the slot into which the argument is to be placed.

If either of B₁ or B₂ is an isomorphism, then both are, and the bilinear form B is said to be nondegenerate. In Mathematics, specifically Linear algebra, a degenerate Bilinear form f(xy on a Vector space V is one such that

If V is finite-dimensional then one can identify V with its double dual V**. One can then show that B₂ is the transpose of the linear map B₁ (if V is infinite-dimensional then B₂ is the transpose of B¹ restricted to the image of V in V**). This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a Given B one can define the transpose of B to be the bilinear form given by

B * (v, w) = B (w, v).

If V is finite-dimensional then the rank of B₁ is equal to the rank of B₂. The column rank of a matrix A is the maximal number of Linearly independent columns of A. If this number is equal to the dimension of V then B₁ and B₂ are linear isomorphisms from V to V*. In this case B is nondegenerate. By the rank-nullity theorem, this is equivalent to the condition that the kernel of B₁ be trivial. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism In fact, for finite dimensional spaces, this is often taken as the definition of nondegeneracy. Thus B is nondegenerate if and only if

$B(v,w)=0\mbox{ for all w}\Rightarrow v=0.$

Given any linear map A : V → V* one can obtain a bilinear form B on V via

B (v, w) = A (v)(w)

This form will be nondegenerate if and only if A is an isomorphism.

Reflexivity and orthogonality

A bilinear form

B : V × V → F

is reflexive if

$B(v,w)=0\Longleftrightarrow B(w,v)=0$ .

Reflexivity allows us to define orthogonality: two vectors v and w are orthogonal with respect to the reflexive bilinear form if and only if :

B (v, w) = 0

B (w, v) = 0

The radical of a bilinear form is the subset of all vectors orthogonal with every other vector. A vector v, with matrix representation x, is in the radical if and only if : $A x= 0 \Longleftrightarrow x^{T} A=0$ The radical is always a subspace of V. It is trivial if and only if the matrix A is nonsingular, and thus if and only if the bilinear form is nondegenerate.

Suppose W is a subspace. Define : $W^{\perp}=\{v| B(v,w)=0\ \forall w\in W\}$

When the bilinear form is nondegenerate, the map $W\leftarrow W^{\perp}$ is bijective, and the dimension of $W^{\perp}$ is dim(V)-dim(W).

One can prove that B is reflexive if and only if it is either:

symmetric : $B (v, w) = B (w, v)$ for all $v,w\in V$ ; or
alternating if $B (v, v) = 0$ for all $v\in V$

Every alternating form is skew-symmetric ( $B (v, w) = - B (w, v)$ ). A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric This may be seen by expanding B(v+w,v+w).

If the characteristic of F is not 2 then the converse is also true (every skew-symmetric form is alternating). In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's If, however, char(F) = 2 then a skew-symmetric form is the same thing as a symmetric form and not all of these are alternating.

A bilinear form is symmetric (resp. skew-symmetric) if and only if its coordinate matrix (relative to any basis) is symmetric (resp. ↔ skew-symmetric). A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when char(F) ≠ 2).

A bilinear form is symmetric if and only if the maps $B_1,B_2\colon V \to V^*$ are equal, and skew-symmetric if and only if they are negatives of one another. If char(F) ≠ 2 then one can always decompose a bilinear form into a symmetric and a skew-symmetric part as follows

$B^{\pm} = \frac{1}{2} (B \pm B^*)$

where B* is the transpose of B (defined above).

Different spaces

Much of the theory is available for a bilinear mapping

B: V × W → F. In Mathematics, a bilinear map is a function of two arguments that is linear in each

In this situation we still have linear mappings of V to the dual space of W, and of W to the dual space of V. It may happen that both of those mappings are isomorphisms; assuming finite dimensions, if one is an isomorphism, the other must be. When this occurs, B is said to be a perfect pairing.

In finite dimensions, this is equivalent to the pairing being nondegenerate (the spaces necessarily having the same dimensions). For modules (instead of vector spaces), nondegenerate is a weaker notion: a pairing can be nondegenerate without being a perfect pairing, for instance $\mathbf{Z} \times \mathbf{Z} \to \mathbf{Z}$ via $(x,y) \mapsto 2xy$ is non-degenerate, but induces multiplication by 2 on the map $\mathbf{Z} \to \mathbf{Z}^*$

Relation to tensor products

By the universal property of the tensor product, bilinear forms on V are in 1-to-1 correspondence with linear maps V ⊗ V → F. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector If B is a bilinear form on V the corresponding linear map is given by

$v\otimes w\mapsto B(v,w).$

The set of all linear maps V ⊗ V → F is the dual space of V ⊗ V, so bilinear forms may be thought of as elements of

$(V\otimes V)^{*} \cong V^{*}\otimes V^{*}.$

Likewise, symmetric bilinear forms may be thought of as elements of S²V* (the second symmetric power of V*), and alternating bilinear forms as elements of Λ²V* (the second exterior power of V*). In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field

On normed vector spaces

A bilinear form on a normed vector space is bounded, if there is a constant $C$ such that for all $u, v\in V$

$B(u,v) \le C \|u\| \|v\|.$

A bilinear form on a normed vector space is elliptic, or coercive, if there is a non-zero constant $c$ such that for all $u\in V$

$B(u,u) \ge c \|u\|^2.$

References

Shilov, Georgi E. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to In Mathematics, a bilinear map is a function of two arguments that is linear in each In Multilinear algebra, a multilinear form is a map of the type f V^N \to K where V is a Vector space In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In Mathematics, a sesquilinear form on a Complex vector space V is a map V × V &rarr C that is linear In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. (1977), Silverman, Richard A. , ed. , Linear Algebra, Dover, ISBN 0-486-63518-X .

External links

Bilinear form on PlanetMath

PlanetMath is a free, collaborative online Mathematics Encyclopedia.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents