Bilinear form
Template:Short description In mathematics, a bilinear form is a bilinear map Template:Math on a vector space Template:Mvar (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function Template:Math that is linear in each argument separately:
- Template:Math Template:Spaces and Template:Spaces Template:Math
- Template:Math Template:Spaces and Template:Spaces Template:Math
The dot product on <math>\R^n</math> is an example of a bilinear form which is also an inner product.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> An example of a bilinear form that is not an inner product would be the four-vector product.
The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms.
When Template:Mvar is the field of complex numbers Template:Math, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.
Coordinate representationEdit
Let Template:Math be an Template:Mvar-dimensional vector space with basis Template:Math.
The Template:Math matrix A, defined by Template:Math is called the matrix of the bilinear form on the basis Template:Math.
If the Template:Math matrix Template:Math represents a vector Template:Math with respect to this basis, and similarly, the Template:Math matrix Template:Math represents another vector Template:Math, then: <math display="block">B(\mathbf{x}, \mathbf{y}) = \mathbf{x}^\textsf{T} A\mathbf{y} = \sum_{i,j=1}^n x_i A_{ij} y_j. </math>
A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are all congruent. More precisely, if Template:Math is another basis of Template:Mvar, then <math display="block">\mathbf{f}_j=\sum_{i=1}^n S_{i,j}\mathbf{e}_i,</math> where the <math>S_{i,j}</math> form an invertible matrix Template:Mvar. Then, the matrix of the bilinear form on the new basis is Template:Math.
PropertiesEdit
Non-degenerate bilinear formsEdit
Template:Further Every bilinear form Template:Math on Template:Mvar defines a pair of linear maps from Template:Mvar to its dual space Template:Math. Define Template:Math by Template:Block indent Template:Block indent This is often denoted as Template:Block indent Template:Block indent where the dot ( ⋅ ) indicates the slot into which the argument for the resulting linear functional is to be placed (see Currying).
For a finite-dimensional vector space Template:Mvar, if either of Template:Math or Template:Math is an isomorphism, then both are, and the bilinear form Template:Math is said to be nondegenerate. More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element:
- <math>B(x,y)=0 </math> for all <math>y \in V</math> implies that Template:Math and
- <math>B(x,y)=0 </math> for all <math>x \in V</math> implies that Template:Math.
The corresponding notion for a module over a commutative ring is that a bilinear form is Template:Visible anchor if Template:Math is an isomorphism. Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular. For example, over the integers, the pairing Template:Math is nondegenerate but not unimodular, as the induced map from Template:Math to Template:Math is multiplication by 2.
If Template:Mvar is finite-dimensional then one can identify Template:Mvar with its double dual Template:Math. One can then show that Template:Math is the transpose of the linear map Template:Math (if Template:Mvar is infinite-dimensional then Template:Math is the transpose of Template:Math restricted to the image of Template:Mvar in Template:Math). Given Template:Math one can define the transpose of Template:Math to be the bilinear form given by Template:Block indent
The left radical and right radical of the form Template:Math are the kernels of Template:Math and Template:Math respectively;Template:Sfn they are the vectors orthogonal to the whole space on the left and on the right.Template:Sfn
If Template:Mvar is finite-dimensional then the rank of Template:Math is equal to the rank of Template:Math. If this number is equal to Template:Math then Template:Math and Template:Math are linear isomorphisms from Template:Mvar to Template:Math. In this case Template:Math is nondegenerate. By the rank–nullity theorem, this is equivalent to the condition that the left and equivalently right radicals be trivial. For finite-dimensional spaces, this is often taken as the definition of nondegeneracy: Template:Block indent
Given any linear map Template:Math one can obtain a bilinear form B on V via Template:Block indent
This form will be nondegenerate if and only if Template:Math is an isomorphism.
If Template:Mvar is finite-dimensional then, relative to some basis for Template:Mvar, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero. Likewise, a nondegenerate form is one for which the determinant of the associated matrix is non-zero (the matrix is non-singular). These statements are independent of the chosen basis. For a module over a commutative ring, a unimodular form is one for which the determinant of the associate matrix is a unit (for example 1), hence the term; note that a form whose matrix determinant is non-zero but not a unit will be nondegenerate but not unimodular, for example Template:Math over the integers.
Symmetric, skew-symmetric, and alternating formsEdit
We define a bilinear form to be
- symmetric if Template:Math for all Template:Math, Template:Math in Template:Mvar;
- alternating if Template:Math for all Template:Math in Template:Mvar;
- Template:Visible anchor or Template:Visible anchor if Template:Math for all Template:Math, Template:Math in Template:Mvar;
- Proposition
- Every alternating form is skew-symmetric.
- Proof
- This can be seen by expanding Template:Math.
If the characteristic of Template:Mvar is not 2 then the converse is also true: every skew-symmetric form is alternating. However, if Template:Math then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetric forms that are not alternating.
A bilinear form is symmetric (respectively skew-symmetric) if and only if its coordinate matrix (relative to any basis) is symmetric (respectively 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 Template:Math).
A bilinear form is symmetric if and only if the maps Template:Math are equal, and skew-symmetric if and only if they are negatives of one another. If Template:Math then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows <math display="block">B^{+} = \tfrac{1}{2} (B + {}^{\text{t}}B) \qquad B^{-} = \tfrac{1}{2} (B - {}^{\text{t}}B) ,</math> where Template:Math is the transpose of Template:Math (defined above).
Reflexive bilinear forms and orthogonal vectorsEdit
Template:Block indent Template:Block indent
A bilinear form Template:Math is reflexive if and only if it is either symmetric or alternating.Template:Sfn In the absence of reflexivity we have to distinguish left and right orthogonality. In a reflexive space the left and right radicals agree and are termed the kernel or the radical of the bilinear form: the subspace of all vectors orthogonal with every other vector. A vector Template:Math, with matrix representation Template:Math, is in the radical of a bilinear form with matrix representation Template:Math, if and only if Template:Math. The radical is always a subspace of Template:Math. It is trivial if and only if the matrix Template:Math is nonsingular, and thus if and only if the bilinear form is nondegenerate.
Suppose Template:Mvar is a subspace. Define the orthogonal complementTemplate:Sfn <math display="block"> W^{\perp} = \left\{\mathbf{v} \mid B(\mathbf{v}, \mathbf{w}) = 0 \text{ for all } \mathbf{w} \in W\right\} .</math>
For a non-degenerate form on a finite-dimensional space, the map Template:Math is bijective, and the dimension of Template:Math is Template:Math.
Bounded and elliptic bilinear formsEdit
Definition: A bilinear form on a normed vector space Template:Math is bounded, if there is a constant Template:Math such that for all Template:Math, <math display="block"> B ( \mathbf{u} , \mathbf{v}) \le C \left\| \mathbf{u} \right\| \left\|\mathbf{v} \right\| .</math>
Definition: A bilinear form on a normed vector space Template:Math is elliptic, or coercive, if there is a constant Template:Math such that for all Template:Math, <math display="block"> B ( \mathbf{u} , \mathbf{u}) \ge c \left\| \mathbf{u} \right\| ^2 .</math>
Associated quadratic formEdit
Template:Further For any bilinear form Template:Math, there exists an associated quadratic form Template:Math defined by Template:Math.
When Template:Math, the quadratic form Q is determined by the symmetric part of the bilinear form B and is independent of the antisymmetric part. In this case there is a one-to-one correspondence between the symmetric part of the bilinear form and the quadratic form, and it makes sense to speak of the symmetric bilinear form associated with a quadratic form.
When Template:Math and Template:Math, this correspondence between quadratic forms and symmetric bilinear forms breaks down.
Relation to tensor productsEdit
By the universal property of the tensor product, there is a canonical correspondence between bilinear forms on Template:Mvar and linear maps Template:Math. If Template:Math is a bilinear form on Template:Mvar the corresponding linear map is given by Template:Block indent In the other direction, if Template:Math is a linear map the corresponding bilinear form is given by composing F with the bilinear map Template:Math that sends Template:Math to Template:Math.
The set of all linear maps Template:Math is the dual space of Template:Math, so bilinear forms may be thought of as elements of Template:Math which (when Template:Mvar is finite-dimensional) is canonically isomorphic to Template:Math.
Likewise, symmetric bilinear forms may be thought of as elements of Template:Math (dual of the second symmetric power of Template:Math) and alternating bilinear forms as elements of Template:Math (the second exterior power of Template:Math). If Template:Math, Template:Math.
GeneralizationsEdit
Pairs of distinct vector spacesEdit
Much of the theory is available for a bilinear mapping from two vector spaces over the same base field to that field Template:Block indent
Here we still have induced linear mappings from Template:Mvar to Template:Math, and from Template:Mvar to Template:Math. It may happen that these 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), just as how a nondegenerate form is weaker than a unimodular form, a nondegenerate pairing is a weaker notion than a perfect pairing. A pairing can be nondegenerate without being a perfect pairing, for instance Template:Math via Template:Math is nondegenerate, but induces multiplication by 2 on the map Template:Math.
Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product".Template:Sfn To define them he uses diagonal matrices Aij having only +1 or −1 for non-zero elements. Some of the "inner products" are symplectic forms and some are sesquilinear forms or Hermitian forms. Rather than a general field Template:Mvar, the instances with real numbers Template:Math, complex numbers Template:Math, and quaternions Template:Math are spelled out. The bilinear form <math display="block">\sum_{k=1}^p x_k y_k - \sum_{k=p+1}^n x_k y_k </math> is called the real symmetric case and labeled Template:Math, where Template:Math. Then he articulates the connection to traditional terminology:Template:Sfn Template:Quote
General modulesEdit
Given a ring Template:Mvar and a right [[Module (mathematics)|Template:Mvar-module]] Template:Math and its dual module Template:Math, a mapping Template:Math is called a bilinear form if Template:Block indent Template:Block indent Template:Block indent for all Template:Math, all Template:Math and all Template:Math.
The mapping Template:Math is known as the natural pairing, also called the canonical bilinear form on Template:Math.Template:Sfn
A linear map Template:Math induces the bilinear form Template:Math, and a linear map Template:Math induces the bilinear form Template:Math.
Conversely, a bilinear form Template:Math induces the R-linear maps Template:Math and Template:Math. Here, Template:Math denotes the double dual of Template:Math.
See alsoEdit
CitationsEdit
ReferencesEdit
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External linksEdit
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