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| In [[mathematics]], a '''reality structure''' on a [[complex vector space]] ''V'' is a decomposition of ''V'' into two real subspaces, called the [[real part|real]] and [[imaginary part]]s of ''V'':
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| :<math>V = V_\mathbb{R} \oplus i V_\mathbb{R}.</math> | |
| Here ''V''<sub>'''R'''</sub> is a real subspace of ''V'', i.e. a subspace of ''V'' considered as a [[vector space]] over the [[real number]]s. If ''V'' has [[complex dimension]] ''n'' (real dimension 2''n''), then ''V''<sub>'''R'''</sub> must have real dimension ''n''.
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| The '''standard reality structure''' on the vector space <math>\mathbb{C}^n</math> is the decomposition
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| :<math>\mathbb{C}^n = \mathbb{R}^n \oplus i\,\mathbb{R}^n.</math>
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| In the presence of a reality structure, every vector in ''V'' has a real part and an imaginary part, each of which is a vector in ''V''<sub>'''R'''</sub>:
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| :<math>v = \operatorname{Re}\{v\}+i\,\operatorname{Im}\{v\}</math>
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| In this case, the [[complex conjugate]] of a vector ''v'' is defined as follows:
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| :<math>\overline v = \operatorname{Re}\{v\} - i\,\operatorname{Im}\{v\}</math>
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| This map <math>v \mapsto \overline v</math> is an [[antilinear]] [[Involution (mathematics)|involution]], i.e.
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| :<math>\overline{\overline v} = v,\quad \overline{v + w} = \overline{v} + \overline{w},\quad\text{and}\quad
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| \overline{\alpha v} = \overline\alpha \, \overline{v}.</math>
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| Conversely, given an antilinear involution <math>v \mapsto c(v)</math> on a complex vector space ''V'', it is possible to define a reality structure on ''V'' as follows. Let
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| :<math>\operatorname{Re}\{v\}=\frac{1}{2}\left(v + c(v)\right),</math>
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| and define
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| :<math>V_\mathbb{R} = \left\{\operatorname{Re}\{v\} \mid v \in V \right\}.</math> | |
| Then
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| :<math>V = V_\mathbb{R} \oplus i V_\mathbb{R}.</math>
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| This is actually the decomposition of ''V'' as the [[eigenspace]]s of the real [[linear operator]] ''c''. The eigenvalues of ''c'' are +1 and −1, with eigenspaces ''V''<sub>'''R'''</sub> and <math>i</math> ''V''<sub>'''R'''</sub>, respectively. Typically, the operator ''c'' itself, rather than the eigenspace decomposition it entails, is referred to as the '''reality structure''' on ''V''.
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| ==See also==
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| *[[Linear complex structure]]
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| *[[Complexification]]
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| ==References==
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| * {{Citation | last1=Penrose | first1=Roger | author1-link=Roger Penrose | last2=Rindler | first2=Wolfgang | author2-link=Wolfgang Rindler | title=Spinors and space-time. Vol. 2 | publisher=[[Cambridge University Press]] | series=Cambridge Monographs on Mathematical Physics | isbn=978-0-521-25267-6 | id={{MathSciNet | id = 838301}} | year=1986}}
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| [[Category:Linear algebra]]
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| {{Linear-algebra-stub}}
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