Bilinear form

In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (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 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 dot product on ${\displaystyle \mathbb {R} ^{n}}$ is an example of a bilinear form.^[1]

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 be an n-dimensional vector space with basis {e₁, …, e_n}.

The n × n matrix A, defined by A_ij = B(e_i, e_j) is called the matrix of the bilinear form on the basis {e₁, …, e_n}.

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

$${\displaystyle B(\mathbf {x} ,\mathbf {y} )=\mathbf {x} ^{\textsf {T}}A\mathbf {y} =\sum _{i,j=1}^{n}x_{i}A_{ij}y_{j}.}$$

 

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 {f₁, …, f_n} is another basis of V, then

$${\displaystyle \mathbf {f} _{j}=\sum _{i=1}^{n}S_{i,j}\mathbf {e} _{i},}$$

 

where the ${\displaystyle S_{i,j}}$  form an invertible matrix S. Then, the matrix of the bilinear form on the new basis is 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:

${\displaystyle B(x,y)=0}$  for all ${\displaystyle y\in V}$  implies that x = 0 and

${\displaystyle B(x,y)=0}$  for all ${\displaystyle x\in V}$  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;^[2] they are the vectors orthogonal to the whole space on the left and on the right.^[3]

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 determinant 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 or antisymmetric 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 (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 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

$${\displaystyle B^{+}={\tfrac {1}{2}}(B+{}^{\text{t}}B)\qquad B^{-}={\tfrac {1}{2}}(B-{}^{\text{t}}B),}$$

 

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.^[4] 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^[5]

$${\displaystyle W^{\perp }=\left\{\mathbf {v} \mid B(\mathbf {v} ,\mathbf {w} )=0{\text{ for all }}\mathbf {w} \in W\right\}.}$$

 

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".^[6] 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

$${\displaystyle \sum _{k=1}^{p}x_{k}y_{k}-\sum _{k=p+1}^{n}x_{k}y_{k}}$$

 

is called the real symmetric case and labeled R(p, q), where p + q = n. Then he articulates the connection to traditional terminology:^[7]

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, there is a canonical correspondence between bilinear forms on V and linear maps 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 F : V ⊗ V → K is a linear map the corresponding bilinear form is given by composing F with the bilinear map V × V → V ⊗ V that sends (v, w) to v⊗w.

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)^∗ which (when V is finite-dimensional) is canonically isomorphic to 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,

$${\displaystyle B(\mathbf {u} ,\mathbf {v} )\leq C\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.}$$

 

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,

$${\displaystyle B(\mathbf {u} ,\mathbf {u} )\geq c\left\|\mathbf {u} \right\|^{2}.}$$

 

Generalization to modules

Given a ring R and a right R-module M and its dual module M^∗, a mapping 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^∗, all x, y ∈ M and all 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.^[8]

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
• Category:Bilinear maps
• Inner product space
• Linear form
• Multilinear form
• Polar space
• Quadratic form
• Sesquilinear form
• System of bilinear equations

Citations

1. "Chapter 3. Bilinear forms — Lecture notes for MA1212" (PDF). 2021-01-16.
2. Jacobson 2009, p. 346.
3. Zhelobenko 2006, p. 11.
4. Grove 1997.
5. Adkins & Weintraub 1992, p. 359.
6. Harvey 1990, p. 22.
7. Harvey 1990, p. 23.
8. Bourbaki 1970, p. 233.

References

• Adkins, William A.; Weintraub, Steven H. (1992), Algebra: An Approach via Module Theory, Graduate Texts in Mathematics, vol. 136, Springer-Verlag, ISBN 3-540-97839-9, Zbl 0768.00003
• 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
• Popov, V. L. (1987), "Bilinear form", in Hazewinkel, M. (ed.), Encyclopedia of Mathematics, vol. 1, Kluwer Academic Publishers, pp. 390–392. Also: Bilinear form, p. 390, at Google Books
• Jacobson, Nathan (2009), Basic Algebra, vol. I (2nd ed.), ISBN 978-0-486-47189-1
• Milnor, J.; Husemoller, D. (1973), Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73, Springer-Verlag, ISBN 3-540-06009-X, Zbl 0292.10016
• Porteous, Ian R. (1995), Clifford Algebras and the Classical Groups, Cambridge Studies in Advanced Mathematics, vol. 50, Cambridge University Press, ISBN 978-0-521-55177-9
• Shafarevich, I. R.; A. O. Remizov (2012), Linear Algebra and Geometry, Springer, ISBN 978-3-642-30993-9
• 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

 

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• "Bilinear form", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Bilinear form". PlanetMath.

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