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| In [[mathematics]], a [[complex number|complex]] [[Matrix_(mathematics)#Square_matrices|square]] [[matrix (mathematics)|matrix]] ''U'' is '''unitary''' if
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| ::<math>U^* U = UU^* = I \,</math>
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| where ''I'' is the [[identity matrix]] and ''U''* is the [[conjugate transpose]] of ''U''.
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| The real analogue of a unitary matrix is an [[orthonormal]] matrix.
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| ==Properties==
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| For any unitary matrix ''U'', the following hold:
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| *Given two complex vectors ''x'' and ''y'', multiplication by ''U'' preserves their [[inner product]]; that is,
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| :<math>\langle Ux, Uy \rangle = \langle x, y \rangle</math>.
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| *''U'' is [[normal matrix|normal]]
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| *''U'' is [[diagonalizable matrix|diagonalizable]]; that is, ''U'' is [[similar matrix|unitarily similar]] to a diagonal matrix, as a consequence of the [[spectral theorem]]. Thus ''U'' has a decomposition of the form
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| ::<math>U = VDV^*\;</math>
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| :where ''V'' is unitary and ''D'' is diagonal and unitary.
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| * <math>|\det(U)|=1</math>.
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| * Its [[Eigenvector#Eigenspaces_of_a_matrix|eigenspaces]] are orthogonal.
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| * For any positive [[integer]] ''n'', the set of all ''n'' by ''n'' unitary matrices with matrix multiplication forms a [[group (mathematics)|group]], called the [[unitary group]] ''U(n)''.
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| * Any square matrix with unit Euclidean norm is the average of two unitary matrices.<ref>{{cite journal| first1=Chi-Kwong|last1= Li |first2= Edward|last2= Poon|doi=10.1080/03081080290025507|title=Additive Decomposition of Real Matrices| year=2002| journal=Linear and Multilinear Algebra| volume=50| issue=4| pages=321–326}}</ref>
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| ==Equivalent conditions==
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| If ''U'' is a square, complex matrix, then the following conditions are equivalent:
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| #''U'' is unitary
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| #''U''* is unitary
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| #''U'' is invertible, with ''U''<sup> –1</sup>=''U''*.
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| # the columns of ''U'' form an [[orthonormal basis]] of <math>\mathbb{C}^n</math> with respect to the usual inner product
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| # the rows of ''U'' form an orthonormal basis of <math>\mathbb{C}^n</math> with respect to the usual inner product
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| # ''U'' is an [[isometry]] with respect to the usual norm
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| # ''U'' is a [[normal matrix]] with [[eigenvalues]] lying on the [[unit circle]].
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| ==See also==
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| * [[Orthogonal matrix]]
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| * [[Hermitian matrix]]
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| * [[Symplectic matrix]]
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| * [[Unitary group]]
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| * [[Special unitary group]]
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| * [[Unitary operator]]
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| * [[Matrix decomposition]]
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| * [[Identity matrix]]
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| * [[Quantum gate]]
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| == References ==
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| {{reflist}}
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| == External links ==
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| * {{MathWorld|urlname=UnitaryMatrix |title=Unitary Matrix |last=Rowland|first= Todd}}
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| * {{SpringerEOM|id=U/u095540|title=Unitary matrix |first=O. A. |last=Ivanova}}
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| {{DEFAULTSORT:Unitary Matrix}}
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| [[Category:Matrices]]
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| [[Category:Unitary operators]]
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Hello from Switzerland. I'm glad to be here. My first name is Rena.
I live in a small city called Egetswil in nothern Switzerland.
I was also born in Egetswil 29 years ago. Married in January 1999. I'm working at the university.
my web site Fifa 15 Coin Generator