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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.
 
==Definition and conventions==
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}}.
 
Specifically a map &phi; : ''V'' &times; ''V'' → '''C''' is sesquilinear if
:<math>\begin{align}
&\phi(x + y, z + w) = \phi(x, z) + \phi(x, w) + \phi(y, z) + \phi(y, w)\\
&\phi(a x, b y) = \bar a b\,\phi(x,y)\end{align}</math>
for all ''x,y,z,w'' &isin; ''V'' and all ''a'', ''b'' &isin; '''C'''. <math>\bar a</math> is the complex conjugate of ''a''.
 
A sesquilinear form can also be viewed as a complex [[bilinear map]]
:<math>\bar V \times V \to \mathbf{C} </math>
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
:<math>\bar V \otimes V \to \mathbf{C}.</math>
 
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''.
 
Given any sesquilinear form &phi; on ''V'' we can define a second sesquilinear form &psi; via the [[conjugate transpose]]:
:<math>\psi(w,z) = \overline{\varphi(z,w)}.</math>
In general, &psi; and &phi; will be different. If they are the same then &phi; is said to be ''Hermitian''. If they are negatives of one another, then &phi; is said to be ''skew-Hermitian''. Every sesquilinear form can be written as a sum of a [[Hermitian form]] and a skew-Hermitian form.
 
== Geometric motivation ==
Bilinear forms are to squaring (''z''<sup>2</sup>), what sesquilinear forms are to [[Euclidean norm]] (|''z''|<sup>2</sup> = ''z''<sup>*</sup>''z'').
 
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.
 
If ''B'' is a bilinear form on a complex vector space and  
<math>|x|_B := B(x,x)</math>  is the associated norm,
then <math>|ix|_B = B(ix,ix) = i^{2}B(x,x) = -|x|_B</math>.
 
By contrast, if ''S'' is a sesquilinear form on a complex vector space and
<math>|x|_S := S(x,x)</math> is the associated norm,  
then <math>|ix|_S = S(ix,ix)=\bar i i S(x,x) = |x|_S</math>.
 
== Hermitian form ==
:''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]].''
 
A '''Hermitian form''' (also called a '''symmetric sesquilinear form'''), is a sesquilinear form ''h'' : ''V'' &times; ''V'' &rarr; '''C''' such that
:<math>h(w,z) = \overline{h(z, w)}.</math>
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
:<math>\langle w,z \rangle = \sum_{i=1}^n \overline{w_i} z_i.</math>
More generally, the [[inner product]] on any complex [[Hilbert space]] is a Hermitian form.
 
A vector space with a Hermitian form (''V'',''h'') is called a '''Hermitian space'''.
 
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''':
:<math>h(w,z) = \overline{\mathbf{w}^T} \mathbf{Hz}. </math>
The components of '''H''' are given by ''H''<sub>''ij''</sub> = ''h''(''e''<sub>''i''</sub>, ''e''<sub>''j''</sub>).
 
The [[quadratic form]] associated to a Hermitian form
:''Q''(''z'') = ''h''(''z'',''z'')
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'' &isin; ''V''.
 
== Skew-Hermitian form ==
A '''skew-Hermitian form''' (also called an '''antisymmetric sesquilinear form'''), is a sesquilinear form &epsilon; : ''V'' &times; ''V'' &rarr; '''C''' such that
:<math>\varepsilon(w,z) = -\overline{\varepsilon(z, w)}.</math>
Every skew-Hermitian form can be written as [[imaginary unit|''i'']] times a Hermitian form.
 
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''':
:<math>\varepsilon(w,z) = \overline{\mathbf{w}}^T \mathbf{Az}.</math>
 
The quadratic form associated to a skew-Hermitian form
:''Q''(''z'') = &epsilon;(''z'',''z'')
is always pure [[imaginary number|imaginary]].
 
==Generalization==
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
 
:A pair consisting of an [[anti-automorphism]] α of the field ''F'' and a function f:''A''×''A''→''F'' satisfying
: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
: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)
:(The "transformation exponential notation" <math>t \mapsto t^{\alpha} \ </math>  is adopted in group theory literature.)
 
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]].
 
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.
 
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.
 
==References==
* K.W. Gruenberg & A.J. Weir (1977) ''Linear Geometry'', §5.8 Sesquilinear Forms, pp 120&ndash;4, Springer, ISBN 0-387-90227-9 .
*{{Springer|id=Sesquilinear_form&oldid=13338|title=Sesquilinear form}}
 
[[Category:Linear algebra]]
[[Category:Functional analysis]]

Revision as of 21:26, 19 February 2014

An internet property portal that features new property launch, resale or rental of personal residential, business and industrial properties in Singapore. Constant and common updates with latest property information, developments and prices.

Flora Ville is a boutique residential development that is steeped in appeal and attraction. Resembling a cosy village, this distinctive residence is made up of fifty unique apartments inside lovely atmosphere of lush landscaping and water options that make every exceptional house a joy to live in. Easy access Singapore with main expressway like Central Expressway (CTE) , Tampines Expressway (TPE) and Seletar Expressway (SLE) present quick access to every part of Singapore with a drive to the guts of the city taking solely 20 minutes.

Urban Vista is situated simply less than 1 minute to the Tanah Merah MRT Station, the development is subsequent to a future commercial hub and is simply a station away from the upcoming fourth university and Changi Enterprise Park, and a couple of stations to the Changi Worldwide Airport Bartley Ridge is a Luxurious Condominium Development at Mount Vernon Road. A Joint Enterprise Growth Between Hong Leong Holdings, CDL & TID. Well-designed items of 1 - four bedrooms types, twin key & Penthouse, with every format fastidiously crafted to offer purposeful and spacious interiors The Senai Backyard, 392 units of FREEHOLD Serviced House (Flagship E, Iskandar Malaysia) by KCC Improvement (M) Sdn Bhd Probably 6 Years Rental Assure!! December 13, 2013 by iskandarinsider

To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to determine your needs and suggest related initiatives, both new launch or resale properties that seemingly match your standards. For those who're searching for resale property, comparable to those few years old, or just acquired Momentary Occupation Permit (TOP), you might click on right here here for fast search and submit your shortlisted listings to us, we'll check and name you for viewing.

Because the title of our web site title suggests, we are decide to deliver you as many new launches in Singapore as attainable till you find one that suits your preference. more data The Santorini is one other upcoming new launch which brought to you by MCC Land(Singapore) Pte Ltd. Undertaking is positioned alongside Tampines Aveune 10 and upcoming new highway,street The upcoming Singapore Univerity of Technology and Design (SUTD) as well as a number of local and worldwide schools resembling Anglican High, Temasek Major, Temasek Junior School and United World School are additionally in the vicinity. As well as, City Vista also gives quick access to different elements of Singapore by way of main expressways comparable to PIE, ECP and TPE. The Glades @ Tanah Merah open for sale housing in singapore grant

Zest Carlton – Make it your property in Australia Zest Carlton is a condominium development at Melbourne. Strategically positioned within the bustling heard of Carlton and approximately 3km away from Central Business District (CBD). It takes a couple of minutes to One Balmoral – Extremely Excessive Finish Freehold Condominium To Match Your Standing! One Balmoral is a extremely excessive end freehold condominium located at One Balmoral Street. One Balmoral A growth by Hong Leong Holdings Limited, consisting of 91 models in Sant Ritz close to Potong Pasir MRT is now open for VVIP booking! Sant Ritz has launched on 05 April 2013. Sant Ritz is an upcoming project located simply three minutes stroll away from Potong Pasir MRT. Sant Ritz european impressed structure is designed 14xxpsf view this challenge

Freedman would probably point out that my marinara sauce is not significantly healthy (wine and bacon, in any case, are just foodie forms of salt, sugar and fat) and, serving for serving, must be dearer than $2-per-jar Ragu. He may argue that in a couple of years, Ragu or General Meals or Kraft will offer a pasta sauce that's nutritionally similar to mine, and that I'd be a snob to not buy it. And he may be right. But for now, neither of us can escape the truth that food, like the whole lot else we purchase, is designed to be low-cost to make, to last ceaselessly and to style better than the subsequent product down the shelf. And likewise like all the things else, after you buy it, you are on your own. International Entrepreneurship and Development Index (GEDI).

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