In linear algebra , the sesquilinear form (Latin sesqui = one and a half) is a function that assigns a scalar value to two vectors and which is linear in one of its two arguments and semilinear in the other. A classic example is the through
![{\ displaystyle f ((v_ {1}, \ ldots, v_ {n}), (w_ {1}, \ ldots, w_ {n})) = {\ overline {v}} _ {1} w_ {1 } + \ ldots + {\ overline {v}} _ {n} w_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97ef2a243e921a43c53ebbb6bcfd6b690c1d4ece)
defined mapping , the complex standard scalar product . Here, the dash denotes the complex conjugation .
![{\ displaystyle f \ colon \ mathbb {C} ^ {n} \ times \ mathbb {C} ^ {n} \ to \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bef2ecbff9051905c7bac8c7732342a7297824b)
The two arguments can come from different vector spaces , but they must be based on a common scalar field ; a sesquilinear form is an illustration ; it is a linear form with respect to one argument and a semi- linear form with respect to the other argument. There are different conventions for the order of linear and semi-linear arguments; in physics it is customary to put the semilinear argument first.
![V, W](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a40b0deabeee6e15bff1e3079b601986d8fe337)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![f \ colon V \ times W \ to K](https://wikimedia.org/api/rest_v1/media/math/render/svg/f18e84392788813e4377eb47f29e4e24ebab8ee3)
Above the real numbers , the concept of the sesquilinear form agrees with that of the bilinear form .
definition
Let there be vector spaces over the complex numbers .
An illustration
![S \ colon V \ times W \ to {\ mathbb C}, \ quad (v, w) \ mapsto S (v, w) = \ langle v, w \ rangle](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f022ad237811aa78e7f277ca34451f3167b28b4)
is called sesquilinear form if is semilinear in the first and linear in the second argument, that is
![\ langle v_1 + v_2, w \ rangle = \ langle v_1, w \ rangle + \ langle v_2, w \ rangle](https://wikimedia.org/api/rest_v1/media/math/render/svg/e090f868ce4aac65a6f652c0fcdcec5556bab1a4)
![\ langle \ lambda v, w \ rangle = \ overline \ lambda \; \ langle v, w \ rangle;](https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb8a5fbc4417cee60e18a240166ace833fcef25)
and
![\ langle v, w_1 + w_2 \ rangle = \ langle v, w_1 \ rangle + \ langle v, w_2 \ rangle](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f87784eb638563027777b650f187ed4b2831c49)
![\ langle v, \ lambda w \ rangle = \ lambda \, \ langle v, w \ rangle.](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd09e52d4d4912c74667190e09e0e8721d8c0510)
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 \ mathbb C](https://wikimedia.org/api/rest_v1/media/math/render/svg/11a6d1585381827bdf73529c2a418bc14098567c)
Sometimes linearity in the first and semi-linearity in the second argument is required instead; however, this difference is of a purely formal nature.
This definition can also be generalized to vector spaces over other bodies or modules over a ring as soon as there is an excellent automorphism or at least endomorphism on the basic body or ring
![\ lambda \ mapsto \ overline \ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd00dd3e6d09fdafc5d1246e62b8052263bbe11)
given is. A candidate for such endomorphisms is the Frobenius homomorphism with positive characteristics .
The constant zero mapping is a sesquilinear form, we write . Pointwise sums and scalar multiples of sesquilinear forms are again sesquilinear forms. The set of sesquilinear forms thus forms a vector space.
![S = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae0fc3192b864f55a46749d8a64e7cf7783d04c)
![\ mathbb {C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
Hermitian sesquilinear form
A sesquilinear form is called Hermitian, if
![{\ displaystyle S \ colon V \ times V \ to \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/131f1312217225e583eef71026928bf43efbed9a)
![S (v, w) = \ overline {S (w, v)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e550d277fdfc824f9d15111a055f9c903aaf89eb)
applies. This definition is analogous to the definition of the symmetrical bilinear form . The adjective "Hermitesch" is derived from the mathematician Charles Hermite .
Examples
An inner product over a complex vector space is a sesquilinear form with Hermitian symmetry, i.e. even a Hermitian form , see also Krein space .
polarization
statement
The so-called polarization formula plays an important role
![{\ begin {aligned} 4 \ cdot S (y, x) & = \ sum _ {{k = 0}} ^ {3} {\ mathrm {i}} ^ {k} S (x + {\ mathrm {i }} ^ {k} y, x + {\ mathrm {i}} ^ {k} y) \\ & = S (x + y, x + y) + {\ mathrm {i}} S (x + {\ mathrm {i}} y, x + {\ mathrm {i}} y) -S (xy, xy) - {\ mathrm {i}} S (x - {\ mathrm {i}} y, x - {\ mathrm { i}} y), \ end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e15bbd5ae71becdc3ce97eec67e972df7a69306)
which shows that the shape is already uniquely determined by its values on the diagonal, ie on pairs of the shape .
![\ langle \ xi, \ xi \ rangle](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d85af9dbb910b352c80e5b9a3dbfdfb63d8cfe)
The polarization formula only applies to sesquilinear forms, but not to general bilinear forms.
Special case
A direct consequence of the polarization formula is the fact that the form already disappears when for all .
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![S (x, x) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/0480003cfc20fd924e59f721da1b6f25acbae91e)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
Or to put it another way: if for everyone , then , well .
![S (x, x) = T (x, x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b6410c00a5d475bc4b7379cee19004bbdb9cc22)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![(ST) (x, x) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/a20264971ad9833e9c35b85b25684cc878196512)
![S = T](https://wikimedia.org/api/rest_v1/media/math/render/svg/73323733279d5bdd3b2921cdc69aa40a27601394)
Counterexample
This statement does not apply to general bilinear forms, consequently there cannot be a polarization formula. This can be seen in the following example. Be and sit
![V = W \ cong {\ mathbb {R}} ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c103b074e8a5511a30e3fb0913222f0b5c71e272)
-
.
is apparently bilinear and it applies
to everyone . On the other hand is
.
![S (x, x) = - x_ {1} x_ {2} + x_ {1} x_ {2} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/56fa53a73bdb4ac332b61057fa00aae37fcaeae3)
![x \ in {\ mathbb {R}} ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/544be0ffe09bee02e6ee9b2977bb7cf55a17258a)
![S ((1,0), (0,1)) = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7315344c4deb02827e3c02def1f33fa7172a457)
Inference
Let be a Hilbert space and a bounded linear operator . Then is a bounded sesquilinear form. The narrowness means that (here ). Conversely, it follows from
Fréchet-Riesz's representation theorem that every bounded sesquilinear form determines a bounded operator
such that for all .
![({\ mathcal {H}}, \ langle \ cdot, \ cdot \ rangle)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3965bf28dc09229ad57a101951cfe7f1ead0494d)
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![S (x, y): = \ langle Tx, y \ rangle](https://wikimedia.org/api/rest_v1/media/math/render/svg/35ef0195dd5eeaaf46c4c4a904bc02e6bf7ccfcf)
![| S (x, y) | \ leq C \ | x \ | \ cdot \ | y \ |](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b2a9de026d18256f28214efe1a23578915fc92)
![C = \ | T \ |](https://wikimedia.org/api/rest_v1/media/math/render/svg/7458c750cf731e71014c6d5d55730803ded09fb2)
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![S (x, y) = \ langle Tx, y \ rangle](https://wikimedia.org/api/rest_v1/media/math/render/svg/186f5035c0340ba8ced640bde3b7126c23d04080)
![x, y \ in {\ mathcal {H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf8cc8a9ebd16ce60a7479d2b2e275cb4bb9b6a)
In particular, disappears exactly when disappears. This can also be easily seen directly as follows: if so follows for all , so
. The converse follows immediately from the definition of .
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
![T](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
![S = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae0fc3192b864f55a46749d8a64e7cf7783d04c)
![\ | Tx \ | ^ {2} = S (x, Tx) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba61e6859401da651f8c9afbb3811d98286c0bc)
![x \ in {\ mathcal {H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/344b77495e65c3a013706035940181ea2dfc2b7a)
![T = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6a5b6d0370b358a8d5f3df6d17eeca08d3629b)
![S.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
With the polarization identity it follows that an operator is zero if and only if for all . However, this statement only applies to the basic field of complex numbers ; the condition that T is self-adjoint is also necessary for the real numbers .
![\ langle Tx, x \ rangle = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc4a160212018a1d49629a6e0ba65d7cc115d5f)
![\ mathbb {C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
Sesquilinear forms on modules
The concept of the sesquilinear form can be generalized to any modules , with the complex conjugation being replaced by any anti-automorphism on the underlying, not necessarily commutative ring . Have modules over the same ring and an anti-automorphism . A picture is said to be -Sesquilinearform if for any , and the following conditions apply:
![M, N](https://wikimedia.org/api/rest_v1/media/math/render/svg/30012178e674e69a55fee64e9c221dbdb0d95b9e)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![\ theta](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![\ langle \ cdot, \ cdot \ rangle \ colon M \ times N \ to R](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ac19c06e511c77d371d9c84cf83f4c8253f876)
![\ theta](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af)
![m, m_ {1}, m_ {2} \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/4660313ca74b06c8b8c0e40edd0ddb3a1c43c991)
![n, n_ {1}, n_ {2} \ in N](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfe7a2c34ad691c12461c0a262e83dd2e682b09)
![\ lambda \ in R](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f6400f0d27779e34813ce2b33d79cd637ff2e8)
![\ langle m_ {1} + m_ {2}, n \ rangle = \ langle m_ {1}, n \ rangle + \ langle m_ {2}, n \ rangle](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e57afc2bf7685ce88673d52e2f0a3da91db1991)
![\ langle m, n_ {1} + n_ {2} \ rangle = \ langle m, n_ {1} \ rangle + \ langle m, n_ {2} \ rangle](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa11334f52e5ed581a7c941d88f7c3bf214f4160)
![\ langle \ lambda m, n \ rangle = \ lambda \ langle m, n \ rangle](https://wikimedia.org/api/rest_v1/media/math/render/svg/5344bef0707993f9344e37a0313d65590a6da5c5)
![\ langle m, \ lambda n \ rangle = \ langle m, n \ rangle \ theta (\ lambda)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c602975a1babe4b59a33a6456421deffaec4ba29)
literature
Individual evidence
-
↑ D. Werner: functional analysis 5th, extended edition. Springer, 2004, ISBN 3-540-21381-3 , Korollar V.5.8, p. 236.
-
^ Nicolas Bourbaki : Algèbre (= Éléments de mathématique ). Springer , Berlin 2007, ISBN 3-540-35338-0 , chap. 9 , p. 10 .