bilinear form

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bilinear form
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In mathematics, a bilinear form on a vector space V is a bilinear map {{nowrap|V × V → K}}, where K is the field of scalars. In other words, a bilinear form is a function {{nowrap|B : V × V → K}} that is linear in each argument separately:
* B(u + v, w) = B(u, w) + B(v, w)     and     B(λu, v) = λB(u, v) * B(u, v + w) = B(u, v) + B(u, w)     and     B(u, λv) = λB(u, v)
The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms.When K 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.

Coordinate representation

Let {{nowrap|V ≅ Kn}} be an n-dimensional vector space with basis {{nowrap|{e1, ..., en}.}} Define the {{nowrap|n × n}} matrix A by {{nowrap|1=Aij = B(ei, ej)}}. If the n × 1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, then:
B(mathbf{v}, mathbf{w}) = mathbf{x}^textsf{T} Amathbf{y} = sum_{i,j=1}^n x_i a_{ij} y_j.
Suppose {{nowrap|{f1, ..., fn}{{void}}}} is another basis for V, such that:
[f1, ..., fn] = [e1, ..., en]S
where {{nowrap|S ∈ GL(n, K)}}. Now the new matrix representation for the bilinear form is given by: STAS.

Maps to the dual space

Every bilinear form B on V defines a pair of linear maps from V to its dual space V∗. Define {{nowrap|B1, B2: V → V∗}} by
B1(v)(w) = B(v, w) B2(v)(w) = B(w, v)
This is often denoted as
B1(v) = B(v, â‹…) B2(v) = B(â‹…, v)
where the dot ( â‹… ) indicates the slot into which the argument for the resulting linear functional is to be placed (see Currying).For a finite-dimensional vector space V, if either of B1 or B2 is an isomorphism, then both are, and the bilinear form B 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:
B(x,y)=0, for all y in V implies that {{nowrap|1=x = 0}} and B(x,y)=0, for all x in V implies that {{nowrap|1=y = 0}}.
The corresponding notion for a module over a commutative ring is that a bilinear form is {{visible anchor|unimodular}} if {{nowrap|V → V∗}} 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 {{nowrap|1=B(x, y) = 2xy}} is nondegenerate but not unimodular, as the induced map from {{nowrap|1=V = Z}} to {{nowrap|1=V∗ = Z}} is multiplication by 2.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∗∗). Given B one can define the transpose of B to be the bilinear form given by
tB(v, w) = B(w, v).
The left radical and right radical of the form B are the kernels of B1 and B2 respectively;{{sfn|Jacobson|2009|page=346}} they are the vectors orthogonal to the whole space on the left and on the right.{{sfn|Zhelobenko|2006|page=11}}If V is finite-dimensional then the rank of B1 is equal to the rank of B2. If this number is equal to dim(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 left and equivalently right radicals be trivial. For finite-dimensional spaces, this is often taken as the definition of nondegeneracy:
Definition: B is nondegenerate if {{nowrap|1=B(v, w) = 0}} for all w implies {{nowrap|1=v = 0}}.
Given any linear map {{nowrap|1=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.If V is finite-dimensional then, relative to some basis for V, 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 is non-zero but not a unit will be nondegenerate but not unimodular, for example {{nowrap|1=B(x, y) = 2xy}} over the integers.

Symmetric, skew-symmetric and alternating forms

We define a bilinear form to be
  • symmetric if {{nowrap|1=B(v, w) = B(w, v)}} for all v, w in V;
  • alternating if {{nowrap|1=B(v, v) = 0}} for all v in V;
  • skew-symmetric if {{nowrap|1=B(v, w) = −B(w, v)}} for all v, w in V;
  • : Proposition: Every alternating form is skew-symmetric.
  • : Proof: This can be seen by expanding {{nowrap|B(v + w, v + w)}}.
If the characteristic of K is not 2 then the converse is also true: every skew-symmetric form is alternating. If, however, {{nowrap|1=char(K) = 2}} 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 (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 {{nowrap|char(K) ≠ 2}}).A bilinear form is symmetric if and only if the maps {{nowrap|B1, B2: V → V∗}} are equal, and skew-symmetric if and only if they are negatives of one another. If {{nowrap|char(K) ≠ 2}} then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows
B^{+} = tfrac{1}{2} (B + {}^{text{t}}B) qquad B^{-} = tfrac{1}{2} (B - {}^{text{t}}B) ,
where tB is the transpose of B (defined above).

Derived quadratic form

For any bilinear form {{nowrap|B : V × V → K}}, there exists an associated quadratic form {{nowrap|Q : V → K}} defined by {{nowrap|Q : V → K : v ↦ B(v, v)}}.When {{nowrap|char(K) ≠ 2}}, 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 {{nowrap|1=char(K) = 2}} and {{nowrap|dim V > 1}}, this correspondence between quadratic forms and symmetric bilinear forms breaks down.

Reflexivity and orthogonality

Definition: A bilinear form {{nowrap|B : V × V → K}} is called reflexive if {{nowrap|1=B(v, w) = 0}} implies {{nowrap|1=B(w, v) = 0}} for all v, w in V. Definition: Let {{nowrap|B : V × V → K}} be a reflexive bilinear form. v, w in V are orthogonal with respect to B if {{nowrap|1=B(v, w) = 0}}.
A bilinear form B is reflexive if and only if it is either symmetric or alternating.{{sfn|Grove|1997}} 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 v, with matrix representation x, is in the radical of a bilinear form with matrix representation A, if and only if {{nowrap|1=Ax = 0 ⇔ xTA = 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 the orthogonal complement{{sfn|Adkins|Weintraub|1992|page=359}}
W^{perp}={mathbf{v} mid B(mathbf{v}, mathbf{w})=0 forall mathbf{w}in W} .
For a non-degenerate form on a finite dimensional space, the map {{nowrap|V/W → W⊥}} is bijective, and the dimension of W⊥ is {{nowrap|dim(V) − dim(W)}}.

Different spaces

Much of the theory is available for a bilinear mapping from two vector spaces over the same base field to that field
B : V × W → K.
Here we still have induced linear mappings from V to W∗, and from W to V∗. 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 {{nowrap|Z × Z → Z}} via {{nowrap|(x, y) ↦ 2xy}} is nondegenerate, but induces multiplication by 2 on the map {{nowrap|Z → Z∗}}.Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product".{{sfn|Harvey|1990|page=22}} 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 K, the instances with real numbers R, complex numbers C, and quaternions H are spelled out. The bilinear form
sum_{k=1}^p x_k y_k - sum_{k=p+1}^n x_k y_k
is called the real symmetric case and labeled {{nowrap|R(p, q)}}, where {{nowrap|1=p + q = n}}. Then he articulates the connection to traditional terminology:{{sfn|Harvey|1990|page=23}}
Some of the real symmetric cases are very important. The positive definite case {{nowrap|R(n, 0)}} is called Euclidean space, while the case of a single minus, {{nowrap|R(n−1, 1)}} is called Lorentzian space. If {{nowrap|1=n = 4}}, then Lorentzian space is also called Minkowski space or Minkowski spacetime. The special case {{nowrap|R(p, p)}} will be referred to as the split-case.

Relation to tensor products

By the universal property of the tensor product, there is a canonical correspondence between bilinear forms on V and linear maps {{nowrap|V ⊗ V → K}}. If B is a bilinear form on V the corresponding linear map is given by
v ⊗ w ↦ B(v, w)
In the other direction, if {{nowrap|F : V ⊗ V → K}} is a linear map the corresponding bilinear form is given by composing F with the bilinear map {{nowrap|V × V → V ⊗ V}} that sends {{nowrap|(v, w)}} to {{nowrap|v⊗w}}.The set of all linear maps {{nowrap|V ⊗ V → K}} is the dual space of {{nowrap|V ⊗ V}}, so bilinear forms may be thought of as elements of {{nowrap|(V ⊗ V)∗}} which (when V is finite-dimensional) is canonically isomorphic to {{nowrap|V∗ ⊗ V∗}}.Likewise, symmetric bilinear forms may be thought of as elements of Sym2(V∗) (the second symmetric power of V∗), and alternating bilinear forms as elements of Λ2V∗ (the second exterior power of V∗).

On normed vector spaces

Definition: A bilinear form on a normed vector space {{nowrap|(V, ‖·‖)}} is bounded, if there is a constant C such that for all {{nowrap|u, v ∈ V}},
B ( mathbf{u} , mathbf{v}) le C left| mathbf{u} right| left|mathbf{v} right| .
Definition: A bilinear form on a normed vector space {{nowrap|(V, ‖·‖)}} is elliptic, or coercive, if there is a constant {{nowrap|c > 0}} such that for all {{nowrap|u ∈ V}},
B ( mathbf{u} , mathbf{u}) ge c left| mathbf{u} right| ^2 .

Generalization to modules

Given a ring R and a right R-module M and its dual module M∗, a mapping {{nowrap|B : M∗ × M → R}} is called a bilinear form if
B(u + v, x) = B(u, x) + B(v, x) B(u, x + y) = B(u, x) + B(u, y) B(αu, xβ) = αB(u, x)β
for all {{nowrap|u, v ∈ M∗}}, {{nowrap|x, y ∈ M}}, {{nowrap|α, β ∈ R}}.The mapping {{nowrap|⋅,⋅{{rangle}} : M∗ × M → R : (u, x) ↦ u(x)}} is known as the natural pairing, also called the canonical bilinear form on {{nowrap|M∗ × M}}.{{sfn|Bourbaki|1970|page=233}}A linear map {{nowrap|S : M∗ → M∗ : u ↦ S(u)}} induces the bilinear form {{nowrap|B : M∗ × M → R : (u, x) ↦ S(u), x{{rangle}}}}, and a linear map {{nowrap|T : M → M : x ↦ T(x)}} induces the bilinear form {{nowrap|B : M∗ × M → R : (u, x) ↦ u, T(x)){{rangle}}}}.Conversely, a bilinear form {{nowrap|B : M∗ × M → R}} induces the R-linear maps {{nowrap|S : M∗ → M∗ : u ↦ (x ↦ B(u, x))}} and {{nowrap|T′ : M → M∗∗ : x ↦ (u ↦ B(u, x))}}. Here, M∗∗ denotes the double dual of M.

See also




  • {{citation | last1=Adkins | first1=William A. | last2=Weintraub | first2=Steven H. | year=1992 | title=Algebra: An Approach via Module Theory | series=Graduate Texts in Mathematics | volume=136 | publisher=Springer-Verlag | isbn=3-540-97839-9 | zbl=0768.00003 }}
  • {{citation | last=Bourbaki | first=N. | year=1970 | title=Algebra | publisher=Springer |authorlink=Nicolas Bourbaki}}
  • {{citation | last=Cooperstein | first=Bruce | year=2010 | title=Advanced Linear Algebra | chapter=Ch 8: Bilinear Forms and Maps | pages=249–88 | publisher=CRC Press | isbn=978-1-4398-2966-0 }}
  • {{citation | last=Grove | first=Larry C. | year=1997 | title=Groups and characters | publisher=Wiley-Interscience | isbn=978-0-471-16340-4}}
  • {{citation | last1=Halmos | first1=Paul R. | author1-link=Paul R. Halmos | year=1974 | title=Finite-dimensional vector spaces | series=Undergraduate Texts in Mathematics | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90093-3 | zbl=0288.15002 }}
  • {{citation | last1=Harvey | first1=F. Reese | year=1990 | title=Spinors and calibrations | chapter=Chapter 2: The Eight Types of Inner Product Spaces | pages=19–40 | publisher=Academic Press | isbn=0-12-329650-1 }}
  • {{citation | editor=Hazewinkel, M. | year=1988 | journal=Encyclopedia of Mathematics | volume=1 | page=390 | publisher=Kluwer Academic Publishers }}
  • {{citation | last=Jacobson | first=Nathan | year=2009 | title=Basic Algebra | volume=I | edition=2nd | isbn=978-0-486-47189-1 }}
  • {{citation | last1=Milnor | first1=J. | author1-link=John Milnor| first2=D. | last2=Husemoller | year=1973 | title=Symmetric Bilinear Forms | series=Ergebnisse der Mathematik und ihrer Grenzgebiete | volume=73 | publisher=Springer-Verlag | isbn=3-540-06009-X | zbl=0292.10016 }}
  • {{citation | last=Porteous | first=Ian R. | authorlink=Ian R. Porteous | year=1995 | title=Clifford Algebras and the Classical Groups | series=Cambridge Studies in Advanced Mathematics | volume=50 | publisher=Cambridge University Press | ISBN=978-0-521-55177-9 }}
  • {{citation | last=Shafarevich | first=I. R. | authorlink=Igor Shafarevich | author2=A. O. Remizov | year=2012 | title = Linear Algebra and Geometry | publisher=Springer | url= | isbn=978-3-642-30993-9}}
  • {{citation | last=Shilov | first=Georgi E. | title=Linear Algebra | editor-last=Silverman | editor-first=Richard A. | year=1977 | publisher=Dover | isbn=0-486-63518-X}}
  • {{citation | last=Zhelobenko | first=DmitriÄ­ Petrovich | year=2006 | title=Principal Structures and Methods of Representation Theory | series=Translations of Mathematical Monographs | publisher=American Mathematical Society | isbn=0-8218-3731-1 }}

External links

  • {{springer|title=Bilinear form|id=p/b016250}}
  • {{planetmath reference|id=1612|title=Bilinear form}}
{{Functional Analysis}}{{PlanetMath attribution|id=7553|title=Unimodular}}

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