Unitary Operator - Examples

Examples

• The identity function is trivially a unitary operator.

• Rotations in R2 are the simplest nontrivial example of unitary operators. Rotations do not change the length of a vector or the angle between 2 vectors. This example can be expanded to R3.

• On the vector space C of complex numbers, multiplication by a number of absolute value 1, that is, a number of the form ei θ for θ ∈ R, is a unitary operator. θ is referred to as a phase, and this multiplication is referred to as multiplication by a phase. Notice that the value of θ modulo 2π does not affect the result of the multiplication, and so the independent unitary operators on C are parametrized by a circle. The corresponding group, which, as a set, is the circle, is called U(1).

• More generally, unitary matrices are precisely the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalization of the notion of a unitary matrix. Orthogonal matrices are the special case of unitary matrices in which all entries are real. They are the unitary operators on Rn.

• The bilateral shift on the sequence space indexed by the integers is unitary. In general, any operator in a Hilbert space which acts by shuffling around an orthonormal basis is unitary. In the finite dimensional case, such operators are the permutation matrices. The unilateral shift is an isometry; its conjugate is a coisometry.

• The Fourier operator is a unitary operator, i.e. the operator which performs the Fourier transform (with proper normalization). This follows from Parseval's theorem.

• Unitary operators are used in unitary representations.