# Bilinear form

In linear algebra, a bilinear form is a function which assigns a scalar value to two vectors and which is linear in both of its arguments.

The two arguments can come from different vector spaces , but they must be based on a common scalar field ; a bilinear form is a map . A bilinear form is a linear form with regard to its first and its second argument, and thus in particular a multilinear form with two arguments. ${\ displaystyle V, W}$ ${\ displaystyle K}$${\ displaystyle B \ colon V \ times W \ to K}$

## definition

There are vector spaces over a body (or more generally a left module and a right module over a not necessarily commutative ring ). ${\ displaystyle V, W}$ ${\ displaystyle K}$ ${\ displaystyle V}$${\ displaystyle W}$

An illustration

${\ displaystyle B \ colon V \ times W \ to K, \ quad (v, w) \ mapsto B (v, w) = \ langle v, w \ rangle}$

is called bilinear form if the two conditions of a linear mapping (additivity and homogeneity) hold in both arguments:

• ${\ displaystyle \ langle v_ {1} + v_ {2}, w \ rangle = \ langle v_ {1}, w \ rangle + \ langle v_ {2}, w \ rangle}$,
• ${\ displaystyle \ langle v, w_ {1} + w_ {2} \ rangle = \ langle v, w_ {1} \ rangle + \ langle v, w_ {2} \ rangle}$,
• ${\ displaystyle \ langle \ lambda v, w \ rangle = \ lambda \ langle v, w \ rangle}$,
• ${\ displaystyle \ langle v, w \ lambda \ rangle = \ langle v, w \ rangle \ lambda}$.

There are , and . ${\ displaystyle v, v_ {1}, v_ {2} \ in V}$${\ displaystyle w, w_ {1}, w_ {2} \ in W}$${\ displaystyle \ lambda \ in K}$

## Symmetry properties in the case V = W

If both arguments of the bilinear form come from the same vector space , it is called the form value of the vector (with respect to ). The bilinear form can have additional symmetry properties: ${\ displaystyle V}$${\ displaystyle B (x, x), x \ in V}$${\ displaystyle x}$${\ displaystyle B}$${\ displaystyle B \ colon V \ times V \ to K}$

• A bilinear form is called symmetric if${\ displaystyle B}$
${\ displaystyle B (x, y) = B (y, x)}$
applies to all .${\ displaystyle x, y \ in V}$
For a symmetrical bilinear form, ( polarization formula ). It follows that the bilinear is determined entirely by the entirety of the form of values, if the underlying field is a characteristic of unequal has .${\ displaystyle 2 \ cdot B (x, y) = B (x + y, x + y) -B (x, x) -B (y, y)}$${\ displaystyle K}$${\ displaystyle 2}$${\ displaystyle (\ operatorname {char} (K) \ neq 2)}$
• A bilinear form is called alternating if all form values ​​with respect to vanish, so if${\ displaystyle B}$${\ displaystyle B}$
${\ displaystyle B (x, x) = 0}$
applies to all .${\ displaystyle x \ in V}$
• A bilinear form is called antisymmetric or skew symmetric, if${\ displaystyle B}$
${\ displaystyle B (x, y) = - B (y, x)}$
applies to all .${\ displaystyle x, y \ in V}$

Any alternating bilinear form is also antisymmetric. If , for example, what is fulfilled for and , then the reverse also applies: Every antisymmetric bilinear form is alternating. Looking more generally at modules over any commutative ring, these two terms are equivalent if the target module does not have a 2- torsion . ${\ displaystyle \ operatorname {char} (K) \ neq 2}$${\ displaystyle K = \ mathbb {R}}$${\ displaystyle K = \ mathbb {C}}$

## Examples

• A scalar product on a real vector space is a non-degenerate, symmetrical, positively definite bilinear form.
• A scalar product on a complex vector space is not a bilinear form, but a sesquilinear form . However, if one understands as a real vector space, then${\ displaystyle B}$${\ displaystyle V}$${\ displaystyle V}$
${\ displaystyle V \ times V \ to \ mathbb {R}, \ quad (x, y) \ mapsto \ mathrm {Re} \, B (x, y)}$
a symmetrical bilinear shape and
${\ displaystyle V \ times V \ to \ mathbb {R}, \ quad (x, y) \ mapsto \ mathrm {Im} \, B (x, y)}$
an alternating bilinear form.
• There is a canonical non-degenerate bilinear form
${\ displaystyle V \ times V ^ {*} \ to K, \ quad (v, f) \ mapsto \ langle v, f \ rangle = f (v).}$

## Degeneration space

### Definition of the degeneration area

Be a bilinear form. The amount ${\ displaystyle B \ colon V \ times W \ to K}$

${\ displaystyle ^ {\ perp} W \ colon = \ left \ {v \ mid \ forall w \ in W \ colon B (v, w) = 0 \ right \} \ subseteq V}$

is a subspace of and is called left kernel or left radical of the bilinear form. The symbolism " " should indicate that elements of the left core are precisely those which (in the sense of the bilinear form) are orthogonal to the entire space . Correspondingly means ${\ displaystyle V}$${\ displaystyle ^ {\ perp} W}$${\ displaystyle W}$

${\ displaystyle V ^ {\ perp} \ colon = \ left \ {w \ mid \ forall v \ in V \ colon B (v, w) = 0 \ right \} \ subseteq W}$

Right-wing core or right-wing radical. If a bilinear form is symmetrical, the right kernel and left kernel match and this space is called the degeneration space of . ${\ displaystyle B \ colon V \ times V \ to K}$${\ displaystyle B}$

The spellings and are also used for subsets or with an analogous definition . ${\ displaystyle R ^ {\ perp}}$${\ displaystyle ^ {\ perp} S}$${\ displaystyle R \ subseteq V}$${\ displaystyle S \ subseteq W}$

### Not degenerate bilinear shape

Each bilinear form defines two linear maps ${\ displaystyle B}$

${\ displaystyle B_ {l} \ colon V \ to W ^ {*}, \ quad v \ mapsto \ left (w \ mapsto B (v, w) \ right)}$

and

${\ displaystyle B_ {r} \ colon W \ to V ^ {*}, \ quad w \ mapsto \ left (v \ mapsto B (v, w) \ right).}$

Right and left kernels are the kernels of these figures:

${\ displaystyle \ ker B_ {l} = {} ^ {\ perp} W}$
${\ displaystyle \ ker B_ {r} = V ^ {\ perp}}$

If both cores trivial (the two images and therefore injective ), it means the bilinear non-degenerate or non-degenerate . Otherwise the bilinear form is called degenerate or degenerate . Are the pictures and even bijective , so isomorphisms , so is the bilinear perfect pairing . This always applies to finite-dimensional vector spaces, the terms not degenerate and perfect can therefore be used synonymously in this case. ${\ displaystyle B_ {l}}$${\ displaystyle B_ {r}}$${\ displaystyle B_ {l}}$${\ displaystyle B_ {r}}$

The bilinear form has not degenerated if the following applies:

• For every vector there is a vector with and${\ displaystyle v \ in V \ setminus \ {0 \}}$${\ displaystyle w \ in W}$${\ displaystyle B (v, w) \ neq 0}$
• for each vector there is a vector with${\ displaystyle w \ in W \ setminus \ {0 \}}$${\ displaystyle v \ in V}$${\ displaystyle B (v, w) \ neq 0.}$

If the bilinear form is symmetrical, then it has not degenerated if and only if its degeneration space is the zero vector space .

## Coordinate representation

For finite-dimensional and one can choose bases and . ${\ displaystyle V}$${\ displaystyle W}$ ${\ displaystyle e = (e_ {1}, \ ldots, e_ {n})}$${\ displaystyle f = (f_ {1}, \ ldots, f_ {m})}$

The representative matrix of a bilinear form is with ${\ displaystyle B \ colon V \ times W \ to K}$${\ displaystyle M_ {B} \ in K ^ {n \ times m}}$

${\ displaystyle {(M_ {B})} _ {ij}: = B (e_ {i}, f_ {j})}$.

If and are the coordinate vectors of and , then applies ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle v \ in V}$${\ displaystyle w \ in W}$

${\ displaystyle B (v, w) = x ^ {T} M_ {B} \, y = {\ begin {pmatrix} x_ {1} \ dots x_ {n} \ end {pmatrix}} {\ begin {pmatrix } B (e_ {1}, f_ {1}) & \ cdots & B (e_ {1}, f_ {m}) \\\ vdots & \ ddots & \ vdots \\ B (e_ {n}, f_ {1 }) & \ dots & B (e_ {n}, f_ {m}) \ end {pmatrix}} {\ begin {pmatrix} y_ {1} \\\ vdots \\ y_ {m} \ end {pmatrix}}}$,

where the matrix product provides a matrix, i.e. a body element. ${\ displaystyle 1 \ times 1}$

Conversely, if any matrix is ​​defined, then ${\ displaystyle M}$${\ displaystyle n \ times m}$

${\ displaystyle B_ {M} (x, y): = x ^ {T} M \, y}$

a bilinear form . ${\ displaystyle B_ {M} \ colon K ^ {n} \ times K ^ {m} \ to K}$

### Change of base

Are and further bases from and , continue the base change matrix from to . Then the matrix of in the new basis results as ${\ displaystyle e '}$${\ displaystyle f '}$${\ displaystyle V}$${\ displaystyle W}$${\ displaystyle {} _ {e '} {\ mathbf {1}} _ {e}}$${\ displaystyle e}$${\ displaystyle e '}$${\ displaystyle B}$

${\ displaystyle A '= {} _ {e} {\ mathbf {1}} _ {e'} ^ {T} \ cdot A \ cdot {} _ {f} {\ mathbf {1}} _ {f ' }}$

If , and , then the matrices and are called congruent to one another . ${\ displaystyle V = W}$${\ displaystyle e = f}$${\ displaystyle e '= f'}$${\ displaystyle A}$${\ displaystyle A '}$

### Examples / properties

• The standard scalar product in has the unit matrix as its matrix with respect to the standard basis .${\ displaystyle \ mathbb {R} ^ {n}}$
• If and the same basis is used for and , then the following applies: The bilinear form is symmetric if and only if the matrix is symmetric , antisymmetric if and only if the matrix is antisymmetric , and alternating if and only if the matrix is alternating .${\ displaystyle V = W}$${\ displaystyle V}$${\ displaystyle W}$
• The figure is a bijection of the space of the bilinear shapes onto the - matrices. If one defines the sum and scalar multiplication of bilinear forms in a canonical way ( ), this bijection is also a vector space isomorphism.${\ displaystyle B \ mapsto M_ {B}}$${\ displaystyle V \ times W \ to K}$${\ displaystyle n \ times m}$${\ displaystyle K}$${\ displaystyle (\ lambda B_ {1} + B_ {2}) (v, w): = \ lambda B_ {1} (v, w) + B_ {2} (v, w)}$
• For symmetrical bilinear forms over vector spaces of finite dimension there is a basis in which the representing matrix has a diagonal shape (if ). (see Gram-Schmidt's orthogonalization method for the special case of positively definite bilinear forms)${\ displaystyle \ operatorname {char} (K) \ neq 2}$
• If further , one can find a basis in which only the entries 1, −1 and 0 appear on the diagonal ( Sylvester's law of inertia )${\ displaystyle K = \ mathbb {R}}$

## Further remarks

• Bilinear forms correspond to linear maps ; see tensor product .${\ displaystyle V \ times W \ to K}$${\ displaystyle V \ otimes W \ to K}$
• If the mapping does not necessarily take place in the scalar body , but in any vector space, one speaks of a bilinear mapping .${\ displaystyle K}$
• The generalization of the concept of the bilinear form to more than two arguments is called the multilinear form .
• Over the field of complex numbers one often demands linearity in one argument and semi- linearity in the other; instead of a bilinear form, a sesquilinear form is obtained . In particular, an inner product over a real vector space is a bilinear form, but over a complex vector space it is only a sesquilinear form.