Bilinear form
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 ≅ K^{n} be an n-dimensional vector space with basis {e_{1}, ..., 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:
Suppose {f_{1}, ..., f_{n}} is another basis for V, such that:
- [f_{1}, ..., f_{n}] = [e_{1}, ..., e_{n}]S
where S ∈ GL(n, K).
Now the new matrix representation for the bilinear form is given by: S^{T}AS.
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 B_{1}, B_{2}: V → 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, ⋅)
- B_{2}(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 B_{1} or B_{2} 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:
- for all implies that x = 0 and
- for all 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 B_{2} is the transpose of the linear map B_{1} (if V is infinite-dimensional then B_{2} is the transpose of B_{1} restricted to the image of V in V^{∗∗}). Given B one can define the transpose of B to be the bilinear form given by
- ^{t}B(v, w) = B(w, v).
The left radical and right radical of the form B are the kernels of B_{1} and B_{2} 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 B_{1} is equal to the rank of B_{2}. If this number is equal to dim(V) then B_{1} and B_{2} 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 B_{1}, B_{2}: 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
where ^{t}B 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^{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 the orthogonal complement^{[4]}
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
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 Sym^{2}(V^{∗}) (the second symmetric power of V^{∗}), and alternating bilinear forms as elements of Λ^{2}V^{∗} (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,
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,
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
Citations
- ↑ Jacobson 2009, p. 346.
- ↑ Zhelobenko 2006, p. 11.
- ↑ Grove 1997.
- ↑ Adkins & Weintraub 1992, p. 359.
- ↑ Harvey 1990, p. 22.
- ↑ Harvey 1990, p. 23.
- ↑ Bourbaki 1970, p. 233.
References
- Adkins, William A.; Weintraub, Steven H. (1992), Algebra: An Approach via Module Theory, Graduate Texts in Mathematics, 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
- Hazewinkel, M., ed. (1988), Encyclopedia of Mathematics, Kluwer Academic Publishers, 1: 390 Missing or empty
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(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
- 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
- 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
- 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
- "Bilinear form". PlanetMath.
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