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.
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![B \ colon V \ times W \ to K](https://wikimedia.org/api/rest_v1/media/math/render/svg/82e56c9967259122d7aef36deb8ab27c430f30db)
definition
There are vector spaces over a body (or more generally a left module and a right module over a not necessarily commutative ring ).
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![W.](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7)
An illustration
![B \ colon V \ times W \ to K, \ quad (v, w) \ mapsto B (v, w) = \ langle v, w \ rangle](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd2157f5aa7cc4e4a1e432bb596b61555a24d3dd)
is called bilinear form if the two conditions of a linear mapping (additivity and homogeneity) hold in both arguments:
-
,
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,
-
,
-
.
There are , and .
![v, v_1, v_2 \ in V](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5cf22b959f54f0f868baf34ab6bdf6633e627de)
![w, w_1, w_2 \ in W](https://wikimedia.org/api/rest_v1/media/math/render/svg/210adc891b0f26e11b248c82111dfd588385d0ba)
![\ lambda \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/22d95fcd9b07168a8162820f7fab4d8ee43366e8)
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:
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![B (x, x), x \ in V](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9c4f6da26ab18f00177a1a2aeceb37f26fbe52)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![B \ colon V \ times V \ to K](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f95198a1e819bf81f8df3be0a4c2b6ae70d683e)
- A bilinear form is called symmetric if
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![B (x, y) = B (y, x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7795bb1aff6bbea03b15e7e7f3c7b8aa019ed71)
- applies to all .
![x, y \ in V](https://wikimedia.org/api/rest_v1/media/math/render/svg/89e1c8106b3df898a0338c2ebce32902e4941fc8)
- 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 .
![2 \ cdot B (x, y) = B (x + y, x + y) -B (x, x) -B (y, y)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cee97d4bace3cb68c42f001c3aa433fa9fbdae5)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![2](https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f)
![(\ operatorname {char} (K) \ neq 2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b858c077538a7a2433892fc8a413294d174490d)
-
A bilinear form is called alternating if all form values with respect to vanish, so if
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![B (x, x) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5672ebc9957580084f45e25ad478bb52e2933c)
- applies to all .
![x \ in V](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa374e20b2db7f6b8caa71ff1865f7f84f215c9f)
- A bilinear form is called antisymmetric or skew symmetric, if
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![B (x, y) = - B (y, x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa77c1ff04be908ca79b9e2430851e8e9b044f52)
- applies to all .
![x, y \ in V](https://wikimedia.org/api/rest_v1/media/math/render/svg/89e1c8106b3df898a0338c2ebce32902e4941fc8)
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 .
![\ operatorname {char} (K) \ neq 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/56504543893e7f42f54b8e30e93c1d9b5c38a4e6)
![K = \ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6419d3aa99701ca996737b17a5e1174d53e6c9e)
![K = \ mathbb C](https://wikimedia.org/api/rest_v1/media/math/render/svg/58665cdd4df26adaa88a248908d1481041a77c9a)
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
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![V \ times V \ to \ mathbb R, \ quad (x, y) \ mapsto \ mathrm {Re} \, B (x, y)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1c160af2162f12740d2b6b11e7eae44037e5113)
- a symmetrical bilinear shape and
![V \ times V \ to \ mathbb R, \ quad (x, y) \ mapsto \ mathrm {Im} \, B (x, y)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a1915919b3f3e7ebdcdcfeab079fb39e30b94e)
- an alternating bilinear form.
- There is a canonical non-degenerate bilinear form
![V \ times V ^ * \ to K, \ quad (v, f) \ mapsto \ langle v, f \ rangle = f (v).](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0f2e7182d65d1e81ad74de319d36752270107f)
Degeneration space
Definition of the degeneration area
Be a bilinear form. The amount
![B \ colon V \ times W \ to K](https://wikimedia.org/api/rest_v1/media/math/render/svg/82e56c9967259122d7aef36deb8ab27c430f30db)
![^ \ perp W \ colon = \ left \ {v \ mid \ forall w \ in W \ colon B (v, w) = 0 \ right \} \ subseteq V](https://wikimedia.org/api/rest_v1/media/math/render/svg/15d6a256eea07e3ca0e609b1983fd8f4f94c5086)
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
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![{\ displaystyle ^ {\ perp} W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7ee0e61c5fea4041554aa320812786accb09e9)
![W.](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7)
![V ^ \ perp \ colon = \ left \ {w \ mid \ forall v \ in V \ colon B (v, w) = 0 \ right \} \ subseteq W](https://wikimedia.org/api/rest_v1/media/math/render/svg/6049e3e8c8b1511717fa27f28635e4bd3171250b)
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 .
![B \ colon V \ times V \ to K](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f95198a1e819bf81f8df3be0a4c2b6ae70d683e)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
The spellings and are also used for subsets or with an analogous definition .
![R ^ \ perp](https://wikimedia.org/api/rest_v1/media/math/render/svg/9784d45eaf020dc36f7b8e54633ac74869389cbd)
![^ \ perp S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6ae7f24227644d1512173b1fdc4efe3ce0797de)
![R \ subseteq V](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2fa9fece99ea256a19d67e51cb3edb9e9a34bb)
![S \ subseteq W](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5366065ad52168d5cf86f41240f996af1de11e5)
Not degenerate bilinear shape
Each bilinear form defines two linear maps
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![B_l \ colon V \ to W ^ *, \ quad v \ mapsto \ left (w \ mapsto B (v, w) \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d378ffd4ee3edecb793a70aa73dd716245b2bcaa)
and
![B_r \ colon W \ to V ^ *, \ quad w \ mapsto \ left (v \ mapsto B (v, w) \ right).](https://wikimedia.org/api/rest_v1/media/math/render/svg/2db0a1a3a524c1774fdfd51a255b7cc74e67c003)
Right and left kernels are the kernels of these figures:
![\ ker B_l = {} ^ \ perp W](https://wikimedia.org/api/rest_v1/media/math/render/svg/2764290603caa4f857c0dfca1b73cba729b0f2d2)
![\ ker B_r = V ^ \ perp](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3283e7cf1051a076f1825f579071fa9173538f7)
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.
![B_l](https://wikimedia.org/api/rest_v1/media/math/render/svg/41713e29b8fcea2b876f662841edbeb3a18bbe45)
![B_r](https://wikimedia.org/api/rest_v1/media/math/render/svg/68e750df8ee313e34c90563be9f828d67c934b55)
![B_l](https://wikimedia.org/api/rest_v1/media/math/render/svg/41713e29b8fcea2b876f662841edbeb3a18bbe45)
![B_r](https://wikimedia.org/api/rest_v1/media/math/render/svg/68e750df8ee313e34c90563be9f828d67c934b55)
The bilinear form has not degenerated if the following applies:
- For every vector there is a vector with and
![v \ in V \ setminus \ {0 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab924be22ccbd82c1de4a218dda58b3105f4db0)
![w \ in W](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0b62771fb4b163c4f1260be1c39768e7173e3e)
![B (v, w) \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/e45262d644051716376ddfbe8368363153ae733a)
- for each vector there is a vector with
![w \ in W \ setminus \ {0 \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/812e89210b8de455df3a1959111da292839f32e0)
![v \ in V](https://wikimedia.org/api/rest_v1/media/math/render/svg/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb)
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 .
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![e = (e_1, \ ldots, e_n)](https://wikimedia.org/api/rest_v1/media/math/render/svg/de86468df1d66583ec2701dcce6f2dd8adb76fa5)
![f = (f_1, \ ldots, f_m)](https://wikimedia.org/api/rest_v1/media/math/render/svg/691a327cf554a674d4d5883b7272971486cad4c2)
The representative matrix of a bilinear form is with
![B \ colon V \ times W \ to K](https://wikimedia.org/api/rest_v1/media/math/render/svg/82e56c9967259122d7aef36deb8ab27c430f30db)
![{\ displaystyle M_ {B} \ in K ^ {n \ times m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a0944d5f5e70928881460c449e7a23b34afe88)
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.
If and are the coordinate vectors of and , then applies
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
![v \ in V](https://wikimedia.org/api/rest_v1/media/math/render/svg/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb)
![w \ in W](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0b62771fb4b163c4f1260be1c39768e7173e3e)
-
,
where the matrix product provides a matrix, i.e. a body element.
![1 \ times 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4bf91a527dc01af9ef6ace81199becf1308e00)
Conversely, if any matrix is defined, then
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
![n \ times m](https://wikimedia.org/api/rest_v1/media/math/render/svg/d82325a2a02ad79bc7c347ba9702ad46eb0de824)
![B_M (x, y): = x ^ TM \, y](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8f1bc13c019809d0e48161484b25d25211994cd)
a bilinear form .
![B_M \ colon K ^ n \ times K ^ m \ to K](https://wikimedia.org/api/rest_v1/media/math/render/svg/679fd718ff2beb5b1ad5d02f28b391ba98e11505)
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
![e '](https://wikimedia.org/api/rest_v1/media/math/render/svg/06c198f6710d781baaf94653df305a4881380033)
![f '](https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![W.](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7)
![{} _ {e '} {\ mathbf 1} _e](https://wikimedia.org/api/rest_v1/media/math/render/svg/7448a0c56b03ddb74d6093cac8fa777e5abf532c)
![e](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467)
![e '](https://wikimedia.org/api/rest_v1/media/math/render/svg/06c198f6710d781baaf94653df305a4881380033)
![B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
![A '= {} _ {e} {\ mathbf 1} _ {e'} ^ T \ cdot A \ cdot {} _ {f} {\ mathbf 1} _ {f '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/529c61d2e2ca6ce9c3f21684875f81f6a1a73f74)
If , and , then the matrices and are called congruent to one another .
![V = W](https://wikimedia.org/api/rest_v1/media/math/render/svg/740038d36bd79466d6938d73b83fe737161fa1c6)
![e = f](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf892cde62554336eeef565d1dce85d7289d79db)
![e '= f'](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cb9f8c0b822a68e2d25b0f3b90789afe822d628)
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![A '](https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6)
Examples / properties
- The standard scalar product in has the unit matrix as its matrix with respect to the standard basis .
![\ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
- 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 .
![V = W](https://wikimedia.org/api/rest_v1/media/math/render/svg/740038d36bd79466d6938d73b83fe737161fa1c6)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![W.](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7)
- 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.
![B \ mapsto M_B](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbcaf2918357bc84b1c08edacdf5f0be37b3610)
![V \ times W \ to K](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a42f846e5bd18c750e24b30c5427ae06d50d70f)
![n \ times m](https://wikimedia.org/api/rest_v1/media/math/render/svg/d82325a2a02ad79bc7c347ba9702ad46eb0de824)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![(\ lambda B_1 + B_2) (v, w): = \ lambda B_1 (v, w) + B_2 (v, w)](https://wikimedia.org/api/rest_v1/media/math/render/svg/2303587ff30b0aefd6ebb8bf6283b5a23546a53b)
- 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)
![\ operatorname {char} (K) \ neq 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/56504543893e7f42f54b8e30e93c1d9b5c38a4e6)
- 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 )
![K = \ mathbb R](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6419d3aa99701ca996737b17a5e1174d53e6c9e)
Further remarks
- Bilinear forms correspond to linear maps ; see tensor product .
![V \ times W \ to K](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a42f846e5bd18c750e24b30c5427ae06d50d70f)
- If the mapping does not necessarily take place in the scalar body , but in any vector space, one speaks of a bilinear mapping .
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
- 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.
Web links
literature