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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>  
where ''I'' is the [[identity matrix]] and ''U''* is the [[conjugate transpose]]  of ''U''.
 
The real analogue of a unitary matrix is an [[orthonormal]] matrix.
 
==Properties==
For any unitary matrix ''U'', the following hold:
*Given two complex vectors ''x'' and ''y'', multiplication by ''U'' preserves their [[inner product]]; that is,
:<math>\langle Ux, Uy \rangle = \langle x, y \rangle</math>.
*''U'' is [[normal matrix|normal]]
*''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
::<math>U = VDV^*\;</math>
:where ''V'' is unitary and ''D'' is diagonal and unitary.
* <math>|\det(U)|=1</math>.
* Its [[Eigenvector#Eigenspaces_of_a_matrix|eigenspaces]] are orthogonal.
* 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)''.
* 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>
 
==Equivalent conditions==
If ''U'' is a square, complex matrix, then the following conditions are equivalent:
#''U'' is unitary
#''U''*  is unitary
#''U'' is invertible, with ''U''<sup> –1</sup>=''U''*.
# the columns of ''U'' form an [[orthonormal basis]] of <math>\mathbb{C}^n</math> with respect to the usual inner product
# the rows of ''U'' form an orthonormal basis of <math>\mathbb{C}^n</math> with respect to the usual inner product
# ''U'' is an [[isometry]] with respect to the usual norm
# ''U'' is a [[normal matrix]] with [[eigenvalues]] lying on the [[unit circle]].
 
==See also==
* [[Orthogonal matrix]]
* [[Hermitian matrix]]
* [[Symplectic matrix]]
* [[Unitary group]]
* [[Special unitary group]]
* [[Unitary operator]]
* [[Matrix decomposition]]
* [[Identity matrix]]
* [[Quantum gate]]
 
== References ==
{{reflist}}
 
== External links ==
* {{MathWorld|urlname=UnitaryMatrix |title=Unitary Matrix |last=Rowland|first= Todd}}
* {{SpringerEOM|id=U/u095540|title=Unitary matrix |first=O. A. |last=Ivanova}}
 
{{DEFAULTSORT:Unitary Matrix}}
[[Category:Matrices]]
[[Category:Unitary operators]]

Revision as of 05:59, 19 February 2014

Psoriasis often also manifests itself on the nails. All types of psoriasis and skin diseases have effective treatment in Ayurveda. In Vanuatu, Tamanu Oil is used by local healers (klevas) in their custom medicine and tropical first aid to treat a wide range of aliments including ALL of the ones listed below and more. And it turns out that these microbes have their favourite spots. All the more that we should feel emphatic and sympathetic for people who have psoriasis because of the difficulty they have to experience day by day.

Even if you are experiencing peeling skin or any of type of psoriasis while taking proteins, vitamins A, vitamins B and vitamins C, it is possible that you can get relief. Thick skin rash (generally identified on the hands, feet, knees, elbows, scalp or lower back areas). Seawater baths are very beneficial in curing psoriasis. You should avoid all commercial versions of herbal skin care products, because their products are not really natural herbs but instead are artificial scents with little bit of essential oils. In the case of psoriasis this process takes place in epidermis and in connective tissue beneath it.

Individuals who scratch one time too often in public are often viewed as people with embarrassing hygiene problems. Case studies have shown that worrying and stress are major factors that aggravate the onset of psoriasis. , so a history of the illness does perform a position. He also observed that the affliction disappears for a quick while only to appear later. Moisturizers maintain skin supple, add water content to the surface of your skin and support you to maintain a youthful look.

For most psoriasis sufferers psoriasis is a very cosmetically disturbing skin disorder. A vastly crucial region of the book is detoxifying and the book supplies a step by step program for creating and applying all-natural home treatments that are able to assist get rid of the symptoms of the skin condition. A new study shows that exercise may help prevent psoriasis. Psoriasis can appear differently in different areas of the skin but it is often small flattened bumps that are raised red or pink patches of skin. Use a solution that is geared to expel all of the makeup from your face.

Few drops of lemon juice or cider vinegar can be added to the rinsing water for a better wash. Sadly 50-80% associated with psoriasis victims can get toe nail psoriasis. In most cases, the person suffering from this condition has got flaky white-silver like patches(scales) and red thick skin. Nearly everyone with psoriasis tries to spend at least a few minutes a day in the sun. If you would like to order the Dermasis Psoriasis cream go here:.

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