Bilinear form

In mathematics, a bilinear form is a bilinear map <span class="texhtml " {{#if: $V \times V \to K$ 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 <span class="texhtml " {{#if: $B : V \times V \to K$ that is linear in each argument separately:

<span class="texhtml " {{#if: $B (u + v, w) = B (u, w) + B (v, w)$ and <span class="texhtml " {{#if: $B (λ u, v) = λB (u, v)$
<span class="texhtml " {{#if: $B (u, v + w) = B (u, v) + B (u, w)$ and <span class="texhtml " {{#if: $B (u, λ v) = λB (u, v)$

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 <span class="texhtml " {{#if: $C$ , one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.

Coordinate representation

Let <span class="texhtml " {{#if: $V$ be an $n$ -dimensional vector space with basis <span class="texhtml " {{#if: ${e 1, \dots, e n}$ .

The <span class="texhtml " {{#if: $n \times n$ matrix A, defined by <span class="texhtml " {{#if: $A ij = B (e i, e j)$ is called the matrix of the bilinear form on the basis <span class="texhtml " {{#if: ${e 1, \dots, e n}$ .

If the <span class="texhtml " {{#if: $n \times 1$ matrix <span class="texhtml " {{#if: $x$ represents a vector <span class="texhtml " {{#if: $x$ with respect to this basis, and similarly, the <span class="texhtml " {{#if: $n \times 1$ matrix <span class="texhtml " {{#if: $y$ represents another vector <span class="texhtml " {{#if: $y$ , 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 <span class="texhtml " {{#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 <span class="texhtml " {{#if:

S T AS

.

Maps to the dual space

Every bilinear form <span class="texhtml " {{#if: $B$ on $V$ defines a pair of linear maps from $V$ to its dual space <span class="texhtml " {{#if: $V *$ . Define <span class="texhtml " {{#if: $B 1, B 2 : V \to V *$ by

<span class="texhtml " {{#if:

B 1 (v)(w) = B (v, w)

<span class="texhtml " {{#if:

B 2 (v)(w) = B (w, v)

This is often denoted as

<span class="texhtml " {{#if:

B 1 (v) = B (v, \cdot)

<span class="texhtml " {{#if:

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 <span class="texhtml " {{#if: $B 1$ or <span class="texhtml " {{#if: $B 2$ is an isomorphism, then both are, and the bilinear form <span class="texhtml " {{#if: $B$ 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 <span class="texhtml " {{#if:

x = 0

and

B(x,y)=0

for all

x\in V

implies that <span class="texhtml " {{#if:

y = 0

.

The corresponding notion for a module over a commutative ring is that a bilinear form is unimodular if <span class="texhtml " {{#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 <span class="texhtml " {{#if: $B (x, y) = 2 xy$ is nondegenerate but not unimodular, as the induced map from <span class="texhtml " {{#if: $V = Z$ to <span class="texhtml " {{#if: $V * = Z$ is multiplication by 2.

If $V$ is finite-dimensional then one can identify $V$ with its double dual <span class="texhtml " {{#if: $V **$ . One can then show that <span class="texhtml " {{#if: $B 2$ is the transpose of the linear map <span class="texhtml " {{#if: $B 1$ (if $V$ is infinite-dimensional then <span class="texhtml " {{#if: $B 2$ is the transpose of <span class="texhtml " {{#if: $B 1$ restricted to the image of $V$ in <span class="texhtml " {{#if: $V **$ ). Given <span class="texhtml " {{#if: $B$ one can define the transpose of <span class="texhtml " {{#if: $B$ to be the bilinear form given by

^tB(v, w) = B(w, v).

The left radical and right radical of the form <span class="texhtml " {{#if: $B$ are the kernels of <span class="texhtml " {{#if: $B 1$ and <span class="texhtml " {{#if: $B 2$ respectively;Lua error: not enough memory.Lua error: not enough memory. they are the vectors orthogonal to the whole space on the left and on the right.Lua error: not enough memory.Lua error: not enough memory.

If $V$ is finite-dimensional then the rank of <span class="texhtml " {{#if: $B 1$ Lua error: not enough memory. is equal to the rank of <span class="texhtml " {{#if: $B 2$ Lua error: not enough memory.. If this number is equal to <span class="texhtml " {{#if: $dim(V)$ Lua error: not enough memory. then <span class="texhtml " {{#if: $B 1$ Lua error: not enough memory. and <span class="texhtml " {{#if: $B 2$ Lua error: not enough memory. are linear isomorphisms from $V$ to <span class="texhtml " {{#if: $V *$ Lua error: not enough memory.. In this case <span class="texhtml " {{#if: $B$ Lua error: not enough memory. 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 <span class="texhtml " {{#if:

B (v, w) = 0

Lua error: not enough memory. for all w implies <span class="texhtml " {{#if:

v = 0

Lua error: not enough memory..

Given any linear map <span class="texhtml " {{#if: $A : V \to V *$ Lua error: not enough memory. 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 <span class="texhtml " {{#if: $A$ Lua error: not enough memory. 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 <span class="texhtml " {{#if: $B (x, y) = 2 xy$ Lua error: not enough memory. over the integers.

Symmetric, skew-symmetric and alternating forms

We define a bilinear form to be

symmetric if <span class="texhtml " {{#if: $B (v, w) = B (w, v)$ Lua error: not enough memory. for all <span class="texhtml " {{#if: $v$ Lua error: not enough memory., <span class="texhtml " {{#if: $w$ Lua error: not enough memory. in $V$ ;
alternating if <span class="texhtml " {{#if: $B (v, v) = 0$ Lua error: not enough memory. for all <span class="texhtml " {{#if: $v$ Lua error: not enough memory. in $V$ ;
skew-symmetric or antisymmetric if <span class="texhtml " {{#if: $B (v, w) = - B (w, v)$ Lua error: not enough memory. for all <span class="texhtml " {{#if: $v$ Lua error: not enough memory., <span class="texhtml " {{#if: $w$ Lua error: not enough memory. in $V$ ;
Proposition

Every alternating form is skew-symmetric.

Proof

This can be seen by expanding <span class="texhtml " {{#if: $B (v + w, v + w)$ Lua error: not enough memory..

If the characteristic of $K$ is not 2 then the converse is also true: every skew-symmetric form is alternating. However, if <span class="texhtml " {{#if: $char(K) = 2$ Lua error: not enough memory. 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 <span class="texhtml " {{#if: $char(K) \neq 2$ Lua error: not enough memory.).

A bilinear form is symmetric if and only if the maps <span class="texhtml " {{#if: $B 1, B 2 : V \to V *$ Lua error: not enough memory. are equal, and skew-symmetric if and only if they are negatives of one another. If <span class="texhtml " {{#if: $char(K) \neq 2$ Lua error: not enough memory. 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 <span class="texhtml " {{#if:

t B

Lua error: not enough memory. is the transpose of <span class="texhtml " {{#if:

B

Lua error: not enough memory. (defined above).

Derived quadratic form

For any bilinear form <span class="texhtml " {{#if: $B : V \times V \to K$ Lua error: not enough memory., there exists an associated quadratic form <span class="texhtml " {{#if: $Q : V \to K$ Lua error: not enough memory. defined by <span class="texhtml " {{#if: $Q : V \to K : v \mapsto B (v, v)$ Lua error: not enough memory..

When <span class="texhtml " {{#if: $char(K) \neq 2$ Lua error: not enough memory., 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 <span class="texhtml " {{#if: $char(K) = 2$ Lua error: not enough memory. and <span class="texhtml " {{#if: $dim V > 1$ Lua error: not enough memory., this correspondence between quadratic forms and symmetric bilinear forms breaks down.

Reflexivity and orthogonality

Definition: A bilinear form <span class="texhtml " {{#if:

B : V \times V \to K

Lua error: not enough memory. is called reflexive if <span class="texhtml " {{#if:

B (v, w) = 0

Lua error: not enough memory. implies <span class="texhtml " {{#if:

B (w, v) = 0

Lua error: not enough memory. for all v, w in V.

Definition: Let <span class="texhtml " {{#if:

B : V \times V \to K

Lua error: not enough memory. be a reflexive bilinear form. v, w in V are orthogonal with respect to B if <span class="texhtml " {{#if:

B (v, w) = 0

Lua error: not enough memory..

A bilinear form <span class="texhtml " {{#if: $B$ Lua error: not enough memory. is reflexive if and only if it is either symmetric or alternating.Lua error: not enough memory.Lua error: not enough memory. 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 <span class="texhtml " {{#if: $v$ Lua error: not enough memory., with matrix representation <span class="texhtml " {{#if: $x$ Lua error: not enough memory., is in the radical of a bilinear form with matrix representation <span class="texhtml " {{#if: $A$ Lua error: not enough memory., if and only if <span class="texhtml " {{#if: $Ax = 0 \Leftrightarrow x T A = 0$ Lua error: not enough memory.. The radical is always a subspace of <span class="texhtml " {{#if: $V$ Lua error: not enough memory.. It is trivial if and only if the matrix <span class="texhtml " {{#if: $A$ Lua error: not enough memory. is nonsingular, and thus if and only if the bilinear form is nondegenerate.

Suppose $W$ is a subspace. Define the orthogonal complementLua error: not enough memory.Lua error: not enough memory.

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 <span class="texhtml " {{#if: $V/W \to W ⊥$ Lua error: not enough memory. is bijective, and the dimension of <span class="texhtml " {{#if: $W ⊥$ Lua error: not enough memory. is <span class="texhtml " {{#if: $dim(V) - dim(W)$ Lua error: not enough memory..

Different spaces

Much of the theory is available for a bilinear mapping from two vector spaces over the same base field to that field

<span class="texhtml " {{#if:

B : V \times W \to K

Lua error: not enough memory..

Here we still have induced linear mappings from $V$ to <span class="texhtml " {{#if: $W *$ Lua error: not enough memory., and from $W$ to <span class="texhtml " {{#if: $V *$ Lua error: not enough memory.. 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 <span class="texhtml " {{#if: $Z \times Z \to Z$ Lua error: not enough memory. via <span class="texhtml " {{#if: $(x, y) \mapsto 2 xy$ Lua error: not enough memory. is nondegenerate, but induces multiplication by 2 on the map <span class="texhtml " {{#if: $Z \to Z *$ Lua error: not enough memory..

Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product".Lua error: not enough memory.Lua error: not enough memory. 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 <span class="texhtml " {{#if: $R$ Lua error: not enough memory., complex numbers <span class="texhtml " {{#if: $C$ Lua error: not enough memory., and quaternions <span class="texhtml " {{#if: $H$ Lua error: not enough memory. 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 <span class="texhtml " {{#if:

R (p, q)

Lua error: not enough memory., where <span class="texhtml " {{#if:

p + q = n

Lua error: not enough memory.. Then he articulates the connection to traditional terminology:Lua error: not enough memory.Lua error: not enough memory.

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.

Lua error: not enough memory.

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 <span class="texhtml " {{#if: $V \otimes V \to K$ Lua error: not enough memory.. If <span class="texhtml " {{#if: $B$ Lua error: not enough memory. is a bilinear form on $V$ the corresponding linear map is given by

<span class="texhtml " {{#if:

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

Lua error: not enough memory.

In the other direction, if <span class="texhtml " {{#if: $F : V \otimes V \to K$ Lua error: not enough memory. is a linear map the corresponding bilinear form is given by composing F with the bilinear map <span class="texhtml " {{#if: $V \times V \to V \otimes V$ Lua error: not enough memory. that sends <span class="texhtml " {{#if: $(v, w)$ Lua error: not enough memory. to <span class="texhtml " {{#if: $v \otimes w$ Lua error: not enough memory..

The set of all linear maps <span class="texhtml " {{#if: $V \otimes V \to K$ Lua error: not enough memory. is the dual space of <span class="texhtml " {{#if: $V \otimes V$ Lua error: not enough memory., so bilinear forms may be thought of as elements of <span class="texhtml " {{#if: $(V \otimes V) *$ Lua error: not enough memory. which (when $V$ is finite-dimensional) is canonically isomorphic to <span class="texhtml " {{#if: $V * \otimes V *$ Lua error: not enough memory..

Likewise, symmetric bilinear forms may be thought of as elements of <span class="texhtml " {{#if: $(Sym 2 V) *$ Lua error: not enough memory. (dual of the second symmetric power of <span class="texhtml " {{#if: $V$ Lua error: not enough memory.) and alternating bilinear forms as elements of <span class="texhtml " {{#if: $(Λ 2 V) * ≃ Λ 2 V *$ Lua error: not enough memory. (the second exterior power of <span class="texhtml " {{#if: $V *$ Lua error: not enough memory.). If <span class="texhtml " {{#if: $char K \neq 2$ Lua error: not enough memory., <span class="texhtml " {{#if: $(Sym 2 V) * ≃ Sym 2 (V *)$ Lua error: not enough memory..

On normed vector spaces

Definition: A bilinear form on a normed vector space <span class="texhtml " {{#if: $(V, ‖\cdot‖)$ Lua error: not enough memory. is bounded, if there is a constant <span class="texhtml " {{#if: $C$ Lua error: not enough memory. such that for all <span class="texhtml " {{#if: $u, v \in V$ Lua error: not enough memory.,

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

Definition: A bilinear form on a normed vector space <span class="texhtml " {{#if: $(V, ‖\cdot‖)$ Lua error: not enough memory. is elliptic, or coercive, if there is a constant <span class="texhtml " {{#if: $c > 0$ Lua error: not enough memory. such that for all <span class="texhtml " {{#if: $u \in V$ Lua error: not enough memory.,

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

Generalization to modules

Given a ring $R$ and a right $R$ -module <span class="texhtml " {{#if: $M$ Lua error: not enough memory. and its dual module <span class="texhtml " {{#if: $M *$ Lua error: not enough memory., a mapping <span class="texhtml " {{#if: $B : M * \times M \to R$ Lua error: not enough memory. is called a bilinear form if

<span class="texhtml " {{#if:

B (u + v, x) = B (u, x) + B (v, x)

Lua error: not enough memory.

<span class="texhtml " {{#if:

B (u, x + y) = B (u, x) + B (u, y)

Lua error: not enough memory.

<span class="texhtml " {{#if:

B (αu, xβ) = αB (u, x) β

Lua error: not enough memory.

for all <span class="texhtml " {{#if: $u, v \in M *$ Lua error: not enough memory., all <span class="texhtml " {{#if: $x, y \in M$ Lua error: not enough memory. and all <span class="texhtml " {{#if: $α, β \in R$ Lua error: not enough memory..

The mapping <span class="texhtml " {{#if: $⟨\cdot,\cdot⟩ : M * \times M \to R : (u, x) \mapsto u (x)$ Lua error: not enough memory. is known as the natural pairing, also called the canonical bilinear form on <span class="texhtml " {{#if: $M * \times M$ Lua error: not enough memory..Lua error: not enough memory.Lua error: not enough memory.

A linear map <span class="texhtml " {{#if: $S : M * \to M * : u \mapsto S (u)$ Lua error: not enough memory. induces the bilinear form <span class="texhtml " {{#if: $B : M * \times M \to R : (u, x) \mapsto ⟨ S (u), x ⟩$ Lua error: not enough memory., and a linear map <span class="texhtml " {{#if: $T : M \to M : x \mapsto T (x)$ Lua error: not enough memory. induces the bilinear form <span class="texhtml " {{#if: $B : M * \times M \to R : (u, x) \mapsto ⟨ u, T (x)⟩$ Lua error: not enough memory..

Conversely, a bilinear form <span class="texhtml " {{#if: $B : M * \times M \to R$ Lua error: not enough memory. induces the R-linear maps <span class="texhtml " {{#if: $S : M * \to M * : u \mapsto (x \mapsto B (u, x))$ Lua error: not enough memory. and <span class="texhtml " {{#if: $T' : M \to M ** : x \mapsto (u \mapsto B (u, x))$ Lua error: not enough memory.. Here, <span class="texhtml " {{#if: $M **$ Lua error: not enough memory. denotes the double dual of <span class="texhtml " {{#if: $M$ Lua error: not enough memory..

Citations

↑ Lua error: not enough memory.

Lua error: not enough memory.

References

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Lua error: not enough memory.. Also: Bilinear form, p. 390, at Google Books
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External links

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Template:Planetmath reference

Template:Functional Analysis Template:PlanetMath attribution

[1] Lua error: not enough memory.

[1]

Coordinate representation

Maps to the dual space

Symmetric, skew-symmetric and alternating forms

Derived quadratic form

Reflexivity and orthogonality

Different spaces

Relation to tensor products

On normed vector spaces

Generalization to modules

See also

Citations

References

External links