Bilinear form

In mathematics, a bilinear form is a bilinear map $V \times V \to K$ 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 $B : V \times V \to K$ that is linear in each argument separately:

$B (u + v, w) = B (u, w) + B (v, w)$ and $B (λ u, v) = λB (u, v)$
$B (u, v + w) = B (u, v) + B (u, w)$ and $B (u, λ v) = λB (u, v)$

The dot product on $\mathbb {R} ^{n}$ 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 ${e 1, \dots, e n}$ .

The $n \times n$ matrix A, defined by $A ij = B (e i, e j)$ is called the matrix of the bilinear form on the basis ${e 1, \dots, e n}$ .

If the $n \times 1$ matrix $x$ represents a vector $x$ with respect to this basis, and similarly, the $n \times 1$ matrix $y$ represents another vector $y$ , then: $B(\mathbf {x} ,\mathbf {y} )=\mathbf {x} ^{\textsf {T}}A\mathbf {y} =\sum _{i,j=1}^{n}x_{i}A_{ij}y_{j}.$

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 ${f 1, \dots, f n}$ is another basis of $V$ , then $\mathbf {f} _{j}=\sum _{i=1}^{n}S_{i,j}\mathbf {e} _{i},$ where the $S_{i,j}$ form an invertible matrix $S$ . Then, the matrix of the bilinear form on the new basis is $S T AS$ .

Properties

Non-degenerate bilinear forms

Every bilinear form $B$ on $V$ defines a pair of linear maps from $V$ to its dual space $V *$ . Define $B 1, B 2 : V \to V *$ 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, \cdot)

B 2 (v) = B (\cdot, 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:

B(x,y)=0

for all

y\in V

implies that

x = 0

and

B(x,y)=0

for all

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 $V \to V *$ 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 $B (x, y) = 2 xy$ 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

^tB(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;^[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 $B 1$ 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 $A : V \to V *$ 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 $B (x, y) = 2 xy$ 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. However, if $char(K) = 2$ 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 $char(K) \neq 2$ ).

A bilinear form is symmetric if and only if the maps $B 1, B 2 : V \to V *$ are equal, and skew-symmetric if and only if they are negatives of one another. If $char(K) \neq 2$ 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 $t B$ is the transpose of $B$ (defined above).

Reflexive bilinear forms and orthogonal vectors

Definition: A bilinear form

B : V \times V \to K

is called reflexive if

B (v, w) = 0

implies

B (w, v) = 0

for all v, w in V.

Definition: Let

B : V \times V \to 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 $Ax = 0 \Leftrightarrow x T A = 0$ . 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^[5] $W^{\perp }=\left\{\mathbf {v} \mid B(\mathbf {v} ,\mathbf {w} )=0{\text{ for all }}\mathbf {w} \in W\right\}.$

For a non-degenerate form on a finite-dimensional space, the map $V/W \to W ⊥$ is bijective, and the dimension of $W ⊥$ is $dim(V) - dim(W)$ .

Bounded and elliptic bilinear forms

Definition: A bilinear form on a normed vector space $(V, ‖\cdot‖)$ is bounded, if there is a constant $C$ such that for all $u, v \in V$ , $B(\mathbf {u} ,\mathbf {v} )\leq C\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.$

Definition: A bilinear form on a normed vector space $(V, ‖\cdot‖)$ is elliptic, or coercive, if there is a constant $c > 0$ such that for all $u \in V$ , $B(\mathbf {u} ,\mathbf {u} )\geq c\left\|\mathbf {u} \right\|^{2}.$

Associated quadratic form

For any bilinear form $B : V \times V \to K$ , there exists an associated quadratic form $Q : V \to K$ defined by $Q : V \to K : v \mapsto B (v, v)$ .

When $char(K) \neq 2$ , 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.

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 $V \otimes V \to K$ . If $B$ is a bilinear form on $V$ the corresponding linear map is given by

v \otimes w \mapsto B (v, w)

In the other direction, if $F : V \otimes V \to K$ is a linear map the corresponding bilinear form is given by composing F with the bilinear map $V \times V \to V \otimes V$ that sends $(v, w)$ to $v \otimes w$ .

The set of all linear maps $V \otimes V \to K$ is the dual space of $V \otimes V$ , so bilinear forms may be thought of as elements of $(V \otimes V) *$ which (when $V$ is finite-dimensional) is canonically isomorphic to $V * \otimes V *$ .

Likewise, symmetric bilinear forms may be thought of as elements of $(Sym 2 V) *$ (dual of the second symmetric power of $V$ ) and alternating bilinear forms as elements of $(Λ 2 V) * ≃ Λ 2 V *$ (the second exterior power of $V *$ ). If $char K \neq 2$ , $(Sym 2 V) * ≃ Sym 2 (V *)$ .

Generalizations

Pairs of distinct vector 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 \times W \to 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 $Z \times Z \to Z$ via $(x, y) \mapsto 2 xy$ is nondegenerate, but induces multiplication by 2 on the map $Z \to Z *$ .

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

General modules

Given a ring $R$ and a right $R$ -module $M$ and its dual module $M *$ , a mapping $B : M * \times M \to R$ 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 $u, v \in M *$ , all $x, y \in M$ and all $α, β \in R$ .

The mapping $⟨\cdot,\cdot⟩ : M * \times M \to R : (u, x) \mapsto u (x)$ is known as the natural pairing, also called the canonical bilinear form on $M * \times M$ .^[8]

A linear map $S : M * \to M * : u \mapsto S (u)$ induces the bilinear form $B : M * \times M \to R : (u, x) \mapsto ⟨ S (u), x ⟩$ , and a linear map $T : M \to M : x \mapsto T (x)$ induces the bilinear form $B : M * \times M \to R : (u, x) \mapsto ⟨ u, T (x)⟩$ .

Conversely, a bilinear form $B : M * \times M \to R$ induces the R-linear maps $S : M * \to M * : u \mapsto (x \mapsto B (u, x))$ and $T' : M \to M ** : x \mapsto (u \mapsto B (u, x))$ . Here, $M **$ denotes the double dual of $M$ .

Citations

^ "Chapter 3. Bilinear forms — Lecture notes for MA1212" (PDF). 2021-01-16.
^ 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, 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.), Courier Corporation, 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

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

This article incorporates material from Unimodular on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

[1] "Chapter 3. Bilinear forms — Lecture notes for MA1212" (PDF). 2021-01-16.

[FOOTNOTEJacobson2009346-2] Jacobson 2009, p. 346.

[FOOTNOTEZhelobenko200611-3] Zhelobenko 2006, p. 11.

[FOOTNOTEGrove1997-4] Grove 1997.

[FOOTNOTEAdkinsWeintraub1992359-5] Adkins & Weintraub 1992, p. 359.

[FOOTNOTEHarvey199022-6] Harvey 1990, p. 22.

[FOOTNOTEHarvey199023-7] Harvey 1990, p. 23.

[FOOTNOTEBourbaki1970233-8] Bourbaki 1970, p. 233.

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