Bilinear form#Symmetric, skew-symmetric and alternating forms

Let {{math|V}} be an {{mvar|n}}-dimensional vector space with basis {{math|{e₁, …, e_n}}}.

The {{math|n × n}} matrix A, defined by {{math|1=A_ij = B(e_i, e_j)}} is called the matrix of the bilinear form on the basis {{math|{e₁, …, e_n}}}.

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

$$B(\backslash mathbf\{x\},\; \backslash mathbf\{y\})\; =\; \backslash mathbf\{x\}^\backslash textsf\{T\}\; A\backslash mathbf\{y\}\; =\; \backslash sum\_\{i,j=1\}^n\; x\_i\; A\_\{ij\}\; y\_j.$$

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

$$\backslash mathbf\{f\}\_j=\backslash sum\_\{i=1\}^n\; S\_\{i,j\}\backslash mathbf\{e\}\_i,$$

where the $S\_\{i,j\}$ form an invertible matrix {{mvar|S}}. Then, the matrix of the bilinear form on the new basis is {{math|S^TAS}}.

{{further|Degenerate bilinear form}}

Every bilinear form {{math|B}} on {{mvar|V}} defines a pair of linear maps from {{mvar|V}} to its dual space {{math|V^∗}}. Define {{math|B₁, B₂: V → V^∗}} by

{{block indent|left=1.6|text={{math|1=B₁(v)(w) = B(v, w)}}}}

{{block indent|left=1.6|text={{math|1=B₂(v)(w) = B(w, v)}}}}

This is often denoted as

{{block indent|left=1.6|text={{math|1=B₁(v) = B(v, ⋅)}}}}

{{block indent|left=1.6|text={{math|1=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 {{mvar|V}}, if either of {{math|B₁}} or {{math|B₂}} is an isomorphism, then both are, and the bilinear form {{math|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 $y\; \backslash in\; V$ implies that {{math|1=x = 0}} and

:$B(x,y)=0$ for all $x\; \backslash in\; V$ implies that {{math|1=y = 0}}.

The corresponding notion for a module over a commutative ring is that a bilinear form is {{visible anchor|unimodular}} if {{math|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 {{math|1=B(x, y) = 2xy}} is nondegenerate but not unimodular, as the induced map from {{math|1=V = Z}} to {{math|1=V^∗ = Z}} is multiplication by 2.

If {{mvar|V}} is finite-dimensional then one can identify {{mvar|V}} with its double dual {{math|V^∗∗}}. One can then show that {{math|B₂}} is the transpose of the linear map {{math|B₁}} (if {{mvar|V}} is infinite-dimensional then {{math|B₂}} is the transpose of {{math|B₁}} restricted to the image of {{mvar|V}} in {{math|1=V^∗∗}}). Given {{math|B}} one can define the transpose of {{math|B}} to be the bilinear form given by

{{block indent|left=1.6|text=^tB(v, w) = B(w, v).}}

The left radical and right radical of the form {{math|B}} are the kernels of {{math|B₁}} and {{math|B₂}} respectively;{{sfn|Jacobson|2009|page=346}} they are the vectors orthogonal to the whole space on the left and on the right.{{sfn|Zhelobenko|2006|page=11}}

If {{mvar|V}} is finite-dimensional then the rank of {{math|B₁}} is equal to the rank of {{math|B₂}}. If this number is equal to {{math|dim(V)}} then {{math|B₁}} and {{math|B₂}} are linear isomorphisms from {{mvar|V}} to {{math|V^∗}}. In this case {{math|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:

{{block indent|left=1.6|text= Definition: B is nondegenerate if {{math|1=B(v, w) = 0}} for all w implies {{math|1=v = 0}}.}}

Given any linear map {{math|1=A : V → V^∗}} one can obtain a bilinear form B on V via

{{block indent|left=1.6|text=B(v, w) = A(v)(w).}}

This form will be nondegenerate if and only if {{math|A}} is an isomorphism.

If {{mvar|V}} is finite-dimensional then, relative to some basis for {{mvar|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 {{math|1=B(x, y) = 2xy}} over the integers.

We define a bilinear form to be

• symmetric if {{math|1=B(v, w) = B(w, v)}} for all {{math|v}}, {{math|w}} in {{mvar|V}};
• alternating if {{math|1= B(v, v) = 0}} for all {{math|v}} in {{mvar|V}};
• {{visible anchor|skew-symmetric bilinear form|text=skew-symmetric}} or {{visible anchor|antisymmetric bilinear form|text=antisymmetric}} if {{math|1=B(v, w) = −B(w, v)}} for all {{math|v}}, {{math|w}} in {{mvar|V}};
• ; Proposition: Every alternating form is skew-symmetric.
• ; Proof: This can be seen by expanding {{math|B(v + w, v + w)}}.

If the characteristic of {{mvar|K}} is not 2 then the converse is also true: every skew-symmetric form is alternating. However, if {{math|1=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 {{math|char(K) ≠ 2}}).

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

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

where {{math|^tB}} is the transpose of {{math|B}} (defined above).

{{block indent|left=1| Definition: A bilinear form {{math|B : V × V → K}} is called reflexive if {{math|1=B(v, w) = 0}} implies {{math|1=B(w, v) = 0}} for all v, w in V.}}

{{block indent|left=1| Definition: Let {{math|B : V × V → K}} be a reflexive bilinear form. v, w in V are orthogonal with respect to B if {{math|1=B(v, w) = 0}}.}}

A bilinear form {{math|B}} is reflexive if and only if it is either symmetric or alternating.{{sfn|Grove|1997}} 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 {{math|v}}, with matrix representation {{math|x}}, is in the radical of a bilinear form with matrix representation {{math|A}}, if and only if {{math|1=Ax = 0 ⇔ x^TA = 0}}. The radical is always a subspace of {{math|V}}. It is trivial if and only if the matrix {{math|A}} is nonsingular, and thus if and only if the bilinear form is nondegenerate.

Suppose {{mvar|W}} is a subspace. Define the orthogonal complement{{sfn|Adkins|Weintraub|1992|page=359}}

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

For a non-degenerate form on a finite-dimensional space, the map {{math|V/W → W^⊥}} is bijective, and the dimension of {{math|W^⊥}} is {{math|dim(V) − dim(W)}}.

Definition: A bilinear form on a normed vector space {{math|(V, ‖⋅‖)}} is bounded, if there is a constant {{math|C}} such that for all {{math|u, v ∈ V}},

$$B\; (\; \backslash mathbf\{u\}\; ,\; \backslash mathbf\{v\})\; \backslash le\; C\; \backslash left\backslash |\; \backslash mathbf\{u\}\; \backslash right\backslash |\; \backslash left\backslash |\backslash mathbf\{v\}\; \backslash right\backslash |\; .$$

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

$$B\; (\; \backslash mathbf\{u\}\; ,\; \backslash mathbf\{u\})\; \backslash ge\; c\; \backslash left\backslash |\; \backslash mathbf\{u\}\; \backslash right\backslash |\; ^2\; .$$

{{further|Quadratic form#Definitions}}

For any bilinear form {{math|B : V × V → K}}, there exists an associated quadratic form {{math|Q : V → K}} defined by {{math|Q : V → K : v ↦ B(v, v)}}.

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

By the universal property of the tensor product, there is a canonical correspondence between bilinear forms on {{mvar|V}} and linear maps {{math|V ⊗ V → K}}. If {{math|B}} is a bilinear form on {{mvar|V}} the corresponding linear map is given by

{{block indent|left=1.6|text= {{math|v ⊗ w ↦ B(v, w)}}}}

In the other direction, if {{math|F : V ⊗ V → K}} is a linear map the corresponding bilinear form is given by composing F with the bilinear map {{math|V × V → V ⊗ V}} that sends {{math|(v, w)}} to {{math|v⊗w}}.

The set of all linear maps {{math|V ⊗ V → K}} is the dual space of {{math|V ⊗ V}}, so bilinear forms may be thought of as elements of {{math|(V ⊗ V)^∗}} which (when {{mvar|V}} is finite-dimensional) is canonically isomorphic to {{math|V^∗ ⊗ V^∗}}.

Likewise, symmetric bilinear forms may be thought of as elements of {{math|(Sym²V)^*}} (dual of the second symmetric power of {{math|V}}) and alternating bilinear forms as elements of {{math|(Λ²V)^∗ ≃ Λ²V^∗}} (the second exterior power of {{math|V^∗}}). If {{math|char(K) ≠ 2}}, {{math|(Sym²V)^* ≃ Sym²(V^∗)}}.

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

{{block indent|left=1.6|text={{math|B : V × W → K}}.}}

Here we still have induced linear mappings from {{mvar|V}} to {{math|W^∗}}, and from {{mvar|W}} to {{math|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 {{math|Z × Z → Z}} via {{math|(x, y) ↦ 2xy}} is nondegenerate, but induces multiplication by 2 on the map {{math|Z → Z^∗}}.

Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product".{{sfn|Harvey|1990|page=22}} 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 {{mvar|K}}, the instances with real numbers {{math|R}}, complex numbers {{math|C}}, and quaternions {{math|H}} are spelled out. The bilinear form

$$\backslash sum\_\{k=1\}^p\; x\_k\; y\_k\; -\; \backslash sum\_\{k=p+1\}^n\; x\_k\; y\_k$$

is called the real symmetric case and labeled {{math|R(p, q)}}, where {{math|1=p + q = n}}. Then he articulates the connection to traditional terminology:{{sfn|Harvey|1990|page=23}}

{{quote|

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

}}

Given a ring {{mvar|R}} and a right Module (mathematics) {{math|M}} and its dual module {{math|M^∗}}, a mapping {{math|B : M^∗ × M → R}} is called a bilinear form if

{{block indent|left=1.6|text={{math|1=B(u + v, x) = B(u, x) + B(v, x)}}}}

{{block indent|left=1.6|text={{math|1=B(u, x + y) = B(u, x) + B(u, y)}}}}

{{block indent|left=1.6|text={{math|1=B(αu, xβ) = αB(u, x)β}}}}

for all {{math|u, v ∈ M^∗}}, all {{math|x, y ∈ M}} and all {{math|α, β ∈ R}}.

The mapping {{math|⟨⋅,⋅⟩ : M^∗ × M → R : (u, x) ↦ u(x)}} is known as the natural pairing, also called the canonical bilinear form on {{math|M^∗ × M}}.{{sfn|Bourbaki|1970|page=233}}

A linear map {{math|S : M^∗ → M^∗ : u ↦ S(u)}} induces the bilinear form {{math|B : M^∗ × M → R : (u, x) ↦ ⟨S(u), x⟩}}, and a linear map {{math|T : M → M : x ↦ T(x)}} induces the bilinear form {{math|B : M^∗ × M → R : (u, x) ↦ ⟨u, T(x)⟩}}.

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

{{cmn|colwidth=20em|

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

}}

{{reflist}}

{{refbegin}}

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{{refend}}

{{Commonscat|Bilinear forms}}

• {{springer|title=Bilinear form|id=p/b016250}}
• {{planetmath reference|urlname=BilinearForm|title=Bilinear form}}

{{Functional Analysis}}

{{PlanetMath attribution|id=7553|title=Unimodular}}

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