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| In [[mathematics]], a '''sesquilinear form''' on a [[complex vector space]] ''V'' is a map ''V'' × ''V'' → '''C''' that is [[linear operator|linear]] in one argument and [[antilinear]] in the other. The name originates from the Latin [[numerical prefix]] [[Wiktionary:sesqui-|''sesqui-'']] meaning "one and a half". Compare with a [[bilinear form]], which is linear in both arguments. However many authors, especially when working solely in a [[complex number|complex]] setting, refer to sesquilinear forms as bilinear forms.
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| A motivating example is the [[inner product]] on a complex vector space, which is not bilinear, but instead sesquilinear. See [[#Geometric motivation|geometric motivation]] below.
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| ==Definition and conventions==
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| Conventions differ as to which argument should be linear. We take the first to be conjugate-linear (i.e. antilinear) and the second to be linear. This is the convention used by essentially all physicists and originates in [[Paul Dirac|Dirac's]] [[bra-ket notation]] in [[quantum mechanics]]. The opposite convention is more common in mathematics{{Citation needed|date=November 2013}}.
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| Specifically a map φ : ''V'' × ''V'' → '''C''' is sesquilinear if
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| :<math>\begin{align}
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| &\phi(x + y, z + w) = \phi(x, z) + \phi(x, w) + \phi(y, z) + \phi(y, w)\\
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| &\phi(a x, b y) = \bar a b\,\phi(x,y)\end{align}</math>
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| for all ''x,y,z,w'' ∈ ''V'' and all ''a'', ''b'' ∈ '''C'''. <math>\bar a</math> is the complex conjugate of ''a''.
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| A sesquilinear form can also be viewed as a complex [[bilinear map]]
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| :<math>\bar V \times V \to \mathbf{C} </math>
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| where <math>\bar V</math> is the [[complex conjugate vector space]] to ''V''. By the universal property of [[tensor product]]s these are in one-to-one correspondence with (complex) linear maps
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| :<math>\bar V \otimes V \to \mathbf{C}.</math>
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| For a fixed ''z'' in ''V'' the map <math>w \mapsto \phi(z,w)</math> is a [[linear functional]] on ''V'' (i.e. an element of the [[dual space]] ''V''*). Likewise, the map <math>w \mapsto \phi(w,z)</math> is a [[conjugate-linear functional]] on ''V''.
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| Given any sesquilinear form φ on ''V'' we can define a second sesquilinear form ψ via the [[conjugate transpose]]:
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| :<math>\psi(w,z) = \overline{\varphi(z,w)}.</math>
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| In general, ψ and φ will be different. If they are the same then φ is said to be ''Hermitian''. If they are negatives of one another, then φ is said to be ''skew-Hermitian''. Every sesquilinear form can be written as a sum of a [[Hermitian form]] and a skew-Hermitian form.
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| == Geometric motivation ==
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| Bilinear forms are to squaring (''z''<sup>2</sup>), what sesquilinear forms are to [[Euclidean norm]] (|''z''|<sup>2</sup> = ''z''<sup>*</sup>''z'').
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| The norm associated to a sesquilinear form is invariant under multiplication by the complex circle (complex numbers of unit norm), while the norm associated to a bilinear form is [[equivariant]] (with respect to squaring). Bilinear forms are ''algebraically '' more natural, while sesquilinear forms are ''geometrically'' more natural.
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| If ''B'' is a bilinear form on a complex vector space and
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| <math>|x|_B := B(x,x)</math> is the associated norm,
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| then <math>|ix|_B = B(ix,ix) = i^{2}B(x,x) = -|x|_B</math>.
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| By contrast, if ''S'' is a sesquilinear form on a complex vector space and
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| <math>|x|_S := S(x,x)</math> is the associated norm,
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| then <math>|ix|_S = S(ix,ix)=\bar i i S(x,x) = |x|_S</math>.
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| == Hermitian form ==
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| :''The term '''Hermitian form''' may also refer to a different concept than that explained below: it may refer to a certain [[differential form]] on a [[Hermitian manifold]].''
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| A '''Hermitian form''' (also called a '''symmetric sesquilinear form'''), is a sesquilinear form ''h'' : ''V'' × ''V'' → '''C''' such that
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| :<math>h(w,z) = \overline{h(z, w)}.</math>
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| The standard Hermitian form on '''C'''<sup>''n''</sup> is given (using again the "physics" convention of linearity in the second and conjugate linearity in the first variable) by
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| :<math>\langle w,z \rangle = \sum_{i=1}^n \overline{w_i} z_i.</math>
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| More generally, the [[inner product]] on any complex [[Hilbert space]] is a Hermitian form.
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| A vector space with a Hermitian form (''V'',''h'') is called a '''Hermitian space'''.
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| If ''V'' is a finite-dimensional space, then relative to any [[basis (linear algebra)|basis]] {''e''<sub>''i''</sub>} of ''V'', a Hermitian form is represented by a [[Hermitian matrix]] '''H''':
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| :<math>h(w,z) = \overline{\mathbf{w}^T} \mathbf{Hz}. </math>
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| The components of '''H''' are given by ''H''<sub>''ij''</sub> = ''h''(''e''<sub>''i''</sub>, ''e''<sub>''j''</sub>).
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| The [[quadratic form]] associated to a Hermitian form
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| :''Q''(''z'') = ''h''(''z'',''z'')
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| is always [[real number|real]]. Actually one can show that a sesquilinear form is Hermitian [[iff]] the associated quadratic form is real for all ''z'' ∈ ''V''.
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| == Skew-Hermitian form ==
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| A '''skew-Hermitian form''' (also called an '''antisymmetric sesquilinear form'''), is a sesquilinear form ε : ''V'' × ''V'' → '''C''' such that
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| :<math>\varepsilon(w,z) = -\overline{\varepsilon(z, w)}.</math>
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| Every skew-Hermitian form can be written as [[imaginary unit|''i'']] times a Hermitian form.
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| If ''V'' is a finite-dimensional space, then relative to any [[basis (linear algebra)|basis]] {''e''<sub>''i''</sub>} of ''V'', a skew-Hermitian form is represented by a [[skew-Hermitian matrix]] '''A''':
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| :<math>\varepsilon(w,z) = \overline{\mathbf{w}}^T \mathbf{Az}.</math>
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| The quadratic form associated to a skew-Hermitian form
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| :''Q''(''z'') = ε(''z'',''z'')
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| is always pure [[imaginary number|imaginary]].
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| ==Generalization==
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| A generalization called a '''semi-bilinear form''' was used by [[Reinhold Baer]] to characterize linear manifolds that are dual to each other in chapter 5 of his book ''Linear Algebra and Projective Geometry'' (1952). For a [[field (mathematics)|field]] ''F'' and ''A'' linear over ''F'' he requires
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| :A pair consisting of an [[anti-automorphism]] α of the field ''F'' and a function f:''A''×''A''→''F'' satisfying
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| :for all ''a,b,c'' ∈ ''A'' <math>f(a+b,c) = f(a,c) + f(b,c),\quad f(a,b+c) = f(a,b) + f(a,c),</math> and
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| :for all ''t'' ∈ ''F'', all ''x,y'' ∈ ''A'' <math>f(t x,y) = t f(x,y),\quad f(x,t y) = f(x,y) t^{\alpha}</math> (page 101)
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| :(The "transformation exponential notation" <math>t \mapsto t^{\alpha} \ </math> is adopted in group theory literature.)
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| Baer calls such a form an α-form over ''A''. The usual sesquilinear form has [[complex conjugation]] for α. When α is the identity, then f is a [[bilinear form]].
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| In the algebraic structure called a [[*-ring]] the anti-automorphism is denoted by * and forms are constructed as indicated for α. Special constructions such as skew-symmetric bilinear forms, Hermitian forms, and skew-Hermitian forms are all considered in the broader context.
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| Particularly in [[L-theory]], one also sees the term '''ε-symmetric''' form, where <math>\epsilon=\pm 1</math>, to refer to both symmetric and skew-symmetric forms.
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| ==References==
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| * K.W. Gruenberg & A.J. Weir (1977) ''Linear Geometry'', §5.8 Sesquilinear Forms, pp 120–4, Springer, ISBN 0-387-90227-9 .
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| *{{Springer|id=Sesquilinear_form&oldid=13338|title=Sesquilinear form}}
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| [[Category:Linear algebra]]
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| [[Category:Functional analysis]]
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