Editing Unitary matrix (section)

==Equivalent conditions==
If ''U'' is a square, complex matrix, then the following conditions are equivalent:<ref>{{Cite book | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher=[[Cambridge University Press]] | isbn=9781139020411 | year=2013 |doi=10.1017/CBO9781139020411 }}</ref>
# <math>U</math> is unitary.
# <math>U^*</math>  is unitary.
# <math>U</math> is invertible with <math>U^{-1} = U^*</math>.
# The columns of <math>U</math> form an [[orthonormal basis]] of <math>\Complex^n</math> with respect to the usual inner product. In other words, <math>U^*U = I</math>. 
# The rows of <math>U</math> form an orthonormal basis of <math>\Complex^n</math> with respect to the usual inner product. In other words, <math>UU^* = I</math>. 
# <math>U</math> is an [[isometry]] with respect to the usual norm. That is, <math>\|Ux\|_2 = \|x\|_2</math> for all <math>x \in \Complex^n</math>, where <math display="inline">\|x\|_2 = \sqrt{\sum_{i=1}^n |x_i|^2}</math>.
# <math>U</math> is a [[normal matrix]] (equivalently, there is an orthonormal basis formed by eigenvectors of <math>U</math>) with eigenvalues lying on the unit circle.