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In mathematics, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function 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 V ≅ Kⁿ be an n-dimensional vector space with basis {e₁, ..., e_n}.

Define the n × n matrix A by A_ij = B(e_i, e_j).

If the n × 1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, then:

B(v,w)=xTAy=∑i,j=1nxiaijyj.{displaystyle B(mathbf {v} ,mathbf {w} )=mathbf {x} ^{mathrm {T} }Amathbf {y} =sum _{i,j=1}^{n}x_{i}a_{ij}y_{j}.}

Suppose {f₁, ..., f_n} is another basis for V, such that:

[f₁, ..., f_n] = [e₁, ..., e_n]S

where S ∈ GL(n, K).

Now the new matrix representation for the bilinear form is given by: S^TAS.

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 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, ⋅)

B₂(v) = B(⋅, v)

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 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{displaystyle B(x,y)=0,} for all y∈V{displaystyle yin V} implies that x = 0 and

B(x,y)=0{displaystyle B(x,y)=0,} for all x∈V{displaystyle xin V} implies that y = 0.

The corresponding notion for a module over a commutative ring is that a bilinear form is unimodular if 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 B(x, y) = 2xy is nondegenerate but not unimodular, as the induced map from V = Z to 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;^[1] they are the vectors orthogonal to the whole space on the left and on the right.^[2]

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 B(v, w) = 0 for all w implies 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.

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 B(x, y) = 2xy over the integers.

Symmetric, skew-symmetric and alternating forms

We define a bilinear form to be

• 
  symmetric if B(v, w) = B(w, v) for all v, w in V;


• 
  alternating if B(v, v) = 0 for all v in V;


• 
  skew-symmetric if 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 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, 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 char(K) ≠ 2).

A bilinear form is symmetric if and only if the maps B₁, B₂: V → V^∗ are equal, and skew-symmetric if and only if they are negatives of one another. If char(K) ≠ 2 then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows

B+=12(B+tB)B−=12(B−tB),{displaystyle 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 B : V × V → K, there exists an associated quadratic form Q : V → K defined by Q : V → K : v ↦ B(v, v).

When 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 char(K) = 2 and dim V > 1, this correspondence between quadratic forms and symmetric bilinear forms breaks down.

Reflexivity and orthogonality

Definition: A bilinear form B : V × V → K is called reflexive if B(v, w) = 0 implies B(w, v) = 0 for all v, w in V.

Definition: Let B : V × V → K be a reflexive bilinear form. v, w in V are orthogonal with respect to B if B(v, w) = 0.

A bilinear form B is reflexive if and only if it is either symmetric or alternating.^[3] 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 Ax = 0 ⇔ x^TA = 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^[4]

W⊥={v∣B(v,w)=0 ∀w∈W} .{displaystyle 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 V/W → W^⊥ is bijective, and the dimension of W^⊥ is 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 Z × Z → Z via (x,y) ↦ 2xy is nondegenerate, but induces multiplication by 2 on the map Z → Z^∗.

Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product".^[5] To define them he uses diagonal matrices A_ij 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

∑k=1pxkyk−∑k=p+1nxkyk{displaystyle 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 R(p, q), where p + q = n. Then he articulates the connection to traditional terminology:^[6]

Some of the real symmetric cases are very important. The positive definite case R(n, 0) is called Euclidean space, while the case of a single minus, R(n−1, 1) is called Lorentzian space. If n = 4, then Lorentzian space is also called Minkowski space or Minkowski spacetime. The special case R(p, p) will be referred to as the split-case.

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 → K. If B is a bilinear form on V a corresponding linear map is given by

v ⊗ w ↦ B(v, w)

Note that this correspondence is by no means unique nor canonical, though.

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

(V ⊗ V)^∗ ≅ 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 (V, ‖·‖) is bounded, if there is a constant C such that for all u, v ∈ V,

B(u,v)≤C‖u‖‖v‖.{displaystyle B(mathbf {u} ,mathbf {v} )leq Cleft|mathbf {u} right|left|mathbf {v} right|.}

Definition: A bilinear form on a normed vector space (V, ‖·‖) is elliptic, or coercive, if there is a constant c > 0 such that for all u ∈ V,

B(u,u)≥c‖u‖2.{displaystyle B(mathbf {u} ,mathbf {u} )geq cleft|mathbf {u} right|^{2}.}

Generalization to modules

Given a ring R and a right R-module M and its dual module M^∗, a mapping 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 u, v ∈ M^∗, x, y ∈ M, α, β ∈ R.

The mapping ⟨⋅,⋅⟩ : M^∗ × M → R : (u, x) ↦ u(x) is known as the natural pairing, also called the canonical bilinear form on M^∗ × M.^[7]

A linear map S : M^∗ → M^∗ : u ↦ S(u) induces the bilinear form B : M^∗ × M → R : (u, x) ↦ ⟨S(u), x⟩, and a linear map T : M → M : x ↦ T(x) induces the bilinear form B : M^∗ × M → R : (u, x) ↦ ⟨u, T(x))⟩.
Conversely, a bilinear form B : M^∗ × M → R induces the R-linear maps S : M^∗ → M^∗ : u ↦ (x ↦ B(u, x)) and T′ : M → M^∗∗ : x ↦ (u ↦ B(u, x)). Here, M^∗∗ denotes the double dual of M.

See also

• Bilinear map


• Bilinear operator


• Inner product space


• Linear form


• Multilinear form


• Quadratic form


• Sesquilinear form


• Polar space

Citations

References

• 
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• 
  Bourbaki, N. (1970), Algebra, Springer
  


• 
  Cooperstein, Bruce (2010), "Ch 8: Bilinear Forms and Maps", Advanced Linear Algebra, CRC Press, pp. 249–88, ISBN 978-1-4398-2966-0
  


• 
  Grove, Larry C. (1997), Groups and characters, Wiley-Interscience, ISBN 978-0-471-16340-4
  


• 
  Halmos, Paul R. (1974), Finite-dimensional vector spaces, Undergraduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90093-3, Zbl 0288.15002
  


• 
  Harvey, F. Reese (1990), "Chapter 2: The Eight Types of Inner Product Spaces", Spinors and calibrations, Academic Press, pp. 19–40, ISBN 0-12-329650-1
  


• 
  Hazewinkel, M., ed. (1988), Encyclopedia of Mathematics, Kluwer Academic Publishers, 1: 390 Missing or empty |title= (help)
  


• 
  Jacobson, Nathan (2009), Basic Algebra, I (2nd ed.), ISBN 978-0-486-47189-1
  


• 
  Milnor, J.; Husemoller, D. (1973), Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, 73, Springer-Verlag, ISBN 3-540-06009-X, Zbl 0292.10016
  


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  Porteous, Ian R. (1995), Clifford Algebras and the Classical Groups, Cambridge Studies in Advanced Mathematics, 50, Cambridge University Press, ISBN 978-0-521-55177-9
  


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  Shafarevich, I. R.; A. O. Remizov (2012), Linear Algebra and Geometry, Springer, ISBN 978-3-642-30993-9
  


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  Shilov, Georgi E. (1977), Silverman, Richard A., ed., Linear Algebra, Dover, ISBN 0-486-63518-X
  


• 
  Zhelobenko, Dmitriĭ Petrovich (2006), Principal Structures and Methods of Representation Theory, Translations of Mathematical Monographs, American Mathematical Society, ISBN 0-8218-3731-1
  

External links

• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Bilinear form", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


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  "Bilinear form". PlanetMath.
  

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