Editing Unitary operator (section)

==Examples==
* The [[identity function]] is trivially a unitary operator.
* [[Rotation|Rotations]] in {{math|'''R'''<sup>2</sup>}} are the simplest nontrivial example of unitary operators. Rotations do not change the length of a vector or the angle between two vectors. This example can be expanded to {{math|'''R'''<sup>3</sup>}}.  In even higher dimensions, this can be extended to the [[Givens rotation]].
* Reflections, like the [[Householder transformation]].
* <math>\frac{1}{\sqrt{n}}</math> times a [[Hadamard matrix]].
* In general, any operator in a Hilbert space that acts by permuting an [[orthonormal basis]] is unitary. In the finite dimensional case, such operators are the [[permutation matrix|permutation matrices]].
* On the [[vector space]] {{math|'''C'''}} of [[complex number]]s, multiplication by a number of [[absolute value]] {{math|1}}, that is, a number of the form {{math|''e<sup>iθ</sup>''}} for {{math|''θ'' ∈ '''R'''}}, is a unitary operator. {{mvar|θ}} is referred to as a phase, and this multiplication is referred to as multiplication by a phase. Notice that the value of {{mvar|θ}} modulo {{math|2''π''}} does not affect the result of the multiplication, and so the independent unitary operators on {{math|'''C'''}} are parametrized by a circle.  The corresponding group, which, as a set, is the circle, is called {{math|[[U(1)]]}}.
* The [[Fourier operator]] is a unitary operator, i.e. the operator that performs the [[Fourier transform]] (with proper normalization). This follows from [[Parseval's theorem]].
* [[Quantum logic gate]]s are unitary operators. Not all gates are [[Self-adjoint operator|Hermitian]].
* More generally, [[unitary matrix|unitary matrices]] are precisely the unitary operators on finite-dimensional [[Hilbert space]]s, so the notion of a unitary operator is a generalization of the notion of a unitary matrix.  [[Orthogonal matrix|Orthogonal matrices]] are the special case of unitary matrices in which all entries are real.<ref>{{harvnb|Roman|2008|loc=p. 238 §10}}</ref> They are the unitary operators on {{math|'''R'''<sup>''n''</sup>}}.
* The [[bilateral shift]] on the [[Lp space|sequence space]] {{math|''ℓ''<sup>2</sup>}} indexed by the [[integer]]s is unitary.
* The [[unilateral shift]] (right shift) is an isometry; its conjugate (left shift) is a coisometry.
* Unitary operators are used in [[unitary representation]]s.
* A '''unitary element''' is a generalization of a unitary operator. In a [[unital algebra]], an element {{mvar|U}} of the algebra is called a unitary element if {{math|''U''*''U'' {{=}} ''UU''* {{=}} ''I''}}, where {{mvar|I}} is the multiplicative [[identity element]].<ref>{{harvnb|Doran|Belfi|1986|p=55}}</ref>
* Any composition of the above.