Editing Unitary matrix (section)

==Properties==
For any unitary matrix {{mvar|U}} of finite size, the following hold:
* Given two complex vectors {{math|'''x'''}} and {{math|'''y'''}}, multiplication by {{mvar|U}} preserves their [[inner product]]; that is, {{math|1=⟨''U'''''x''', ''U'''''y'''⟩ = ⟨'''x''', '''y'''⟩}}.
* {{mvar|U}} is [[normal matrix|normal]] (<math>U^* U = UU^*</math>).
* {{mvar|U}} is [[diagonalizable matrix|diagonalizable]]; that is, {{mvar|U}} is [[similar matrix|unitarily similar]] to a diagonal matrix, as a consequence of the [[spectral theorem]]. Thus, {{mvar|U}} has a decomposition of the form <math>U = VDV^*,</math> where {{mvar|V}} is unitary, and {{mvar|D}} is diagonal and unitary.
* The [[eigenvalues]] of <math>U</math> lie on the [[unit circle]], as does <math>\det(U)</math>.
* The [[eigenspace]]s of <math>U</math> are orthogonal.
* {{mvar|U}} can be written as {{math|1=''U'' = ''e''<sup>''iH''</sup>}}, where {{mvar|e}} indicates the [[matrix exponential]], {{mvar|i}} is the imaginary unit, and {{mvar|H}} is a [[Hermitian matrix]].

For any nonnegative [[integer]] {{math|''n''}}, the set of all {{math|''n'' × ''n''}} unitary matrices with matrix multiplication forms a [[group (mathematics)|group]], called the [[unitary group]] {{math|U(''n'')}}.

Every 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|s2cid= 120125694 }}</ref>