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单词 Unitary
释义

unitary


0.1 Definitions

  • A unitary space V is a complex vector space with adistinguished positive definitePlanetmathPlanetmath Hermitian formMathworldPlanetmathPlanetmath,

    -,-:V×V,

    which serves asthe inner productMathworldPlanetmath on V.

  • A unitary transformation is a surjectivePlanetmathPlanetmath linear transformationT:VV satisfying

    u,v=Tu,Tv,u,vV.(1)

    These are isometriesMathworldPlanetmath of V.

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

    Tv,TuV=v,uU,u,vU

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

  • A unitary matrix is a square complex-valued matrix, A, whose inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathis equal to its conjugate transposeMathworldPlanetmath:

    A-1=A¯t.
  • 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:

    T-1=T*

    In Hilbert spaces unitary transformations correspond precisely to unitary operators.

0.2 Remarks

  1. 1.

    A standard example of a unitary space isn with inner product

    u,v=i=1nuivi¯,u,vn.(2)
  2. 2.

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

  3. 3.

    A unitary transformation is an automorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath. This follows fromthe fact that a unitary transformation T preserves theinner-product norm:

    Tu=u,uV.(3)

    Hence, if

    Tu=0,

    then by the definition (1)it follows that

    u=0,

    and hence by the inner-product axioms that

    u=0.

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

  4. 4.

    Moreover, relationMathworldPlanetmathPlanetmath (3) can be taken as the definitionof a unitary transformation. Indeed, using the polarizationidentityPlanetmathPlanetmath it is possible to show that ifT preserves the norm, then (1) must hold as well.

  5. 5.

    A simple example of a unitary matrix is the change ofcoordinates matrix between two orthonormal bases. Indeed, letu1,,un and v1,,vn be two orthonormal bases, andlet A=(Aji) be the corresponding change of basis matrixdefined by

    vj=iAjiui,j=1,,n.

    Substituting the above relation into the defining relations for anorthonormal basis,

    ui,uj=δij,
    vk,vl=δkl,

    we obtain

    ijδijAkiAlj¯=iAkiAli¯=δkl.

    In matrix notation, the above is simply

    AA¯t=I,

    as desired.

  6. 6.

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

    STu,STv=Tu,Tv=u,v

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

  7. 7.

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

Titleunitary
Canonical nameUnitary
Date of creation2013-03-22 12:02:01
Last modified on2013-03-22 12:02:01
Ownerasteroid (17536)
Last modified byasteroid (17536)
Numerical id21
Authorasteroid (17536)
Entry typeDefinition
Classificationmsc 47D03
Classificationmsc 47B99
Classificationmsc 47A05
Classificationmsc 46C05
Classificationmsc 15-00
Synonymcomplex inner product space
Related topicEuclideanVectorSpace2
Related topicPauliMatrices
Definesunitary space
Definesunitary matrix
Definesunitary transformation
Definesunitary operator
Definesunitary group
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