unitary
0.1 Definitions
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A unitary space is a complex vector space with a distinguished positive definite Hermitian form,
which serves as the inner product on .
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A unitary transformation is a surjective linear transformation satisfying
(1) These are isometries of .
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More generally, a unitary transformation is a surjective linear transformation between two unitary spaces satisfying
In this entry will restrict to the case of the first , i.e. .
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A unitary matrix is a square complex-valued matrix, , whose inverse is equal to its conjugate transpose:
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When is a Hilbert space, a bounded linear operator is said to be a unitary operator if its inverse is equal to its adjoint:
In Hilbert spaces unitary transformations correspond precisely to unitary operators.
0.2 Remarks
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A standard example of a unitary space is with inner product
(2) -
2.
Unitary transformations and unitary matrices are closely related. On the one hand, a unitary matrix defines a unitary transformation of relative to the inner product (2). On the other hand, the representing matrix of a unitary transformation relative to an orthonormal basis is, in fact, a unitary matrix.
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3.
A unitary transformation is an automorphism. This follows from the fact that a unitary transformation preserves the inner-product norm:
(3) Hence, if
then by the definition (1) it follows that
and hence by the inner-product axioms that
Thus, the kernel of is trivial, and therefore it is an automorphism.
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4.
Moreover, relation (3) can be taken as the definition of a unitary transformation. Indeed, using the polarization identity it is possible to show that if preserves the norm, then (1) must hold as well.
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A simple example of a unitary matrix is the change of coordinates matrix between two orthonormal bases. Indeed, let and be two orthonormal bases, and let be the corresponding change of basis matrix defined by
Substituting the above relation into the defining relations for an orthonormal basis,
we obtain
In matrix notation, the above is simply
as desired.
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6.
Unitary transformations form a group under composition. Indeed, if are unitary transformations then is also surjective and
for every . Hence is also a unitary transformation.
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7.
Unitary spaces, transformations, 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 |