0.1 Definitions

  • More generally, a unitary transformation is a surjective linear transformation T:UV between two unitary spaces U,V satisfying


    In this entry will restrict to the case of the first , i.e. U=V.

  • When V is a Hilbert spaceMathworldPlanetmath, a bounded linear operator T:VV is said to be a unitary operator if its inverse is equal to its adjointPlanetmathPlanetmath:


    In Hilbert spaces unitary transformations correspond precisely to unitary operators.

0.2 Remarks

  1. 1.

    A standard example of a unitary space is n with inner product

    u,v=i=1nuivi¯,u,vn. (2)
  2. 2.

    Unitary transformations and unitary matrices are closely related. On the one hand, a unitary matrix defines a unitary transformation of n relative to the inner product (2). On the other hand, the representing matrix of a unitary transformation relative to an orthonormal basisMathworldPlanetmath is, in fact, a unitary matrix.

  3. 3.

    A unitary transformation is an automorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath. This follows from the fact that a unitary transformation T preserves the inner-product norm:

    Tu=u,uV. (3)

    Hence, if


    then by the definition (1) it follows that


    and hence by the inner-product axioms that


    Thus, the kernel of T is trivial, and therefore it is an automorphism.

  4. 4.

    Moreover, relationMathworldPlanetmathPlanetmath (3) can be taken as the definition of a unitary transformation. Indeed, using the polarization identityPlanetmathPlanetmath it is possible to show that if T preserves the norm, then (1) must hold as well.

  5. 5.

    A simple example of a unitary matrix is the change of coordinates matrix between two orthonormal bases. Indeed, let u1,,un and v1,,vn be two orthonormal bases, and let A=(Aji) be the corresponding change of basis matrix defined by


    Substituting the above relation into the defining relations for an orthonormal basis,

    ui,uj = δij,
    vk,vl = δkl,

    we obtain


    In matrix notation, the above is simply


    as desired.

  6. 6.

    Unitary transformations form a group under composition. Indeed, if S,T are unitary transformations then ST is also surjective and


    for every u,vV. Hence ST is also a unitary transformation.

  7. 7.

    Unitary spaces, transformationsMathworldPlanetmath, matrices and operators are of fundamental importance in quantum mechanics.

Title unitary
Canonical name Unitary
Date of creation 2013-03-22 12:02:01
Last modified on 2013-03-22 12:02:01
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 21
Author asteroid (17536)
Entry type Definition
Classification msc 47D03
Classification msc 47B99
Classification msc 47A05
Classification msc 46C05
Classification msc 15-00
Synonym complex inner product space
Related topic EuclideanVectorSpace2
Related topic PauliMatrices
Defines unitary space
Defines unitary matrix
Defines unitary transformation
Defines unitary operator
Defines unitary group