# Sesquilinear form

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

${\ displaystyle f ((v_ {1}, \ ldots, v_ {n}), (w_ {1}, \ ldots, w_ {n})) = {\ overline {v}} _ {1} w_ {1 } + \ ldots + {\ overline {v}} _ {n} w_ {n}}$ defined mapping , the complex standard scalar product . Here, the dash denotes the complex conjugation . ${\ displaystyle f \ colon \ mathbb {C} ^ {n} \ times \ mathbb {C} ^ {n} \ to \ mathbb {C}}$ 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. ${\ displaystyle V, W}$ ${\ displaystyle K}$ ${\ displaystyle f \ colon V \ times W \ to K}$ 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 . ${\ displaystyle V, W}$ An illustration

${\ displaystyle S \ colon V \ times W \ to \ mathbb {C}, \ quad (v, w) \ mapsto S (v, w) = \ langle v, w \ rangle}$ is called sesquilinear form if is semilinear in the first and linear in the second argument, that is ${\ displaystyle S}$ • ${\ displaystyle \ langle v_ {1} + v_ {2}, w \ rangle = \ langle v_ {1}, w \ rangle + \ langle v_ {2}, w \ rangle}$ • ${\ displaystyle \ langle \ lambda v, w \ rangle = {\ overline {\ lambda}} \; \ langle v, w \ rangle;}$ and

• ${\ displaystyle \ langle v, w_ {1} + w_ {2} \ rangle = \ langle v, w_ {1} \ rangle + \ langle v, w_ {2} \ rangle}$ • ${\ displaystyle \ langle v, \ lambda w \ rangle = \ lambda \, \ langle v, w \ rangle.}$ There are , and . ${\ displaystyle v, v_ {1}, v_ {2} \ in V}$ ${\ displaystyle w, w_ {1}, w_ {2} \ in W}$ ${\ displaystyle \ lambda \ in \ mathbb {C}}$ 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

${\ displaystyle \ lambda \ mapsto {\ overline {\ lambda}}}$ 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. ${\ displaystyle S = 0}$ ${\ displaystyle \ mathbb {C}}$ ## Hermitian sesquilinear form

A sesquilinear form is called Hermitian, if ${\ displaystyle S \ colon V \ times V \ to \ mathbb {C}}$ ${\ displaystyle S (v, w) = {\ overline {S (w, v)}}}$ 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

{\ displaystyle {\ 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}}} which shows that the shape is already uniquely determined by its values ​​on the diagonal, ie on pairs of the shape . ${\ displaystyle \ langle \ xi, \ xi \ rangle}$ 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 . ${\ displaystyle S}$ ${\ displaystyle S (x, x) = 0}$ ${\ displaystyle x}$ Or to put it another way: if for everyone , then , well . ${\ displaystyle S (x, x) = T (x, x)}$ ${\ displaystyle x}$ ${\ displaystyle (ST) (x, x) = 0}$ ${\ displaystyle S = T}$ ### 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 ${\ displaystyle V = W \ cong \ mathbb {R} ^ {2}}$ ${\ displaystyle S (x, y): = x ^ {T} {\ begin {pmatrix} 0 & -1 \\ 1 & 0 \ end {pmatrix}} y = -x_ {1} y_ {2} + x_ {2} y_ {1}}$ .

${\ displaystyle S}$ is apparently bilinear and it applies to everyone . On the other hand is . ${\ displaystyle S (x, x) = - x_ {1} x_ {2} + x_ {1} x_ {2} = 0}$ ${\ displaystyle x \ in \ mathbb {R} ^ {2}}$ ${\ displaystyle S ((1,0), (0,1)) = 1}$ ### 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 . ${\ displaystyle ({\ mathcal {H}}, \ langle \ cdot, \ cdot \ rangle)}$ ${\ displaystyle T}$ ${\ displaystyle S (x, y): = \ langle Tx, y \ rangle}$ ${\ displaystyle | S (x, y) | \ leq C \ | x \ | \ cdot \ | y \ |}$ ${\ displaystyle C = \ | T \ |}$ ${\ displaystyle T}$ ${\ displaystyle S (x, y) = \ langle Tx, y \ rangle}$ ${\ displaystyle x, y \ in {\ mathcal {H}}}$ 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 . ${\ displaystyle S}$ ${\ displaystyle T}$ ${\ displaystyle S = 0}$ ${\ displaystyle \ | Tx \ | ^ {2} = S (x, Tx) = 0}$ ${\ displaystyle x \ in {\ mathcal {H}}}$ ${\ displaystyle T = 0}$ ${\ displaystyle S}$ 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 . ${\ displaystyle \ langle Tx, x \ rangle = 0}$ ${\ displaystyle x}$ ${\ displaystyle \ mathbb {C}}$ ## 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: ${\ displaystyle M, N}$ ${\ displaystyle R}$ ${\ displaystyle \ theta}$ ${\ displaystyle R}$ ${\ displaystyle \ langle \ cdot, \ cdot \ rangle \ colon M \ times N \ to R}$ ${\ displaystyle \ theta}$ ${\ displaystyle m, m_ {1}, m_ {2} \ in M}$ ${\ displaystyle n, n_ {1}, n_ {2} \ in N}$ ${\ displaystyle \ lambda \ in R}$ • ${\ displaystyle \ langle m_ {1} + m_ {2}, n \ rangle = \ langle m_ {1}, n \ rangle + \ langle m_ {2}, n \ rangle}$ • ${\ displaystyle \ langle m, n_ {1} + n_ {2} \ rangle = \ langle m, n_ {1} \ rangle + \ langle m, n_ {2} \ rangle}$ • ${\ displaystyle \ langle \ lambda m, n \ rangle = \ lambda \ langle m, n \ rangle}$ • ${\ displaystyle \ langle m, \ lambda n \ rangle = \ langle m, n \ rangle \ theta (\ lambda)}$ ## Individual evidence

1. D. Werner: functional analysis 5th, extended edition. Springer, 2004, ISBN 3-540-21381-3 , Korollar V.5.8, p. 236.
2. ^ Nicolas Bourbaki : Algèbre (=  Éléments de mathématique ). Springer , Berlin 2007, ISBN 3-540-35338-0 , chap. 9 , p. 10 .