unitary
0.1 Definitions
- •
A unitary space is a complex vector space with adistinguished positive definite
Hermitian form
,
which serves asthe inner product
on .
- •
A unitary transformation is a surjective
linear transformation satisfying
(1) These are isometries
of .
- •
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. .
- •
A unitary matrix is a square complex-valued matrix, , whose inverse
is equal to its conjugate transpose
:
- •
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
- 1.
A standard example of a unitary space is with inner product
(2) - 2.
Unitary transformations and unitary matricesare closely related. On the one hand, a unitary matrix defines aunitary transformation of relative to the inner product(2). On the other hand, the representing matrix of aunitary transformation relative to an orthonormal basis
is, in fact, aunitary matrix.
- 3.
A unitary transformation is an automorphism
. This follows fromthe fact that a unitary transformation preserves theinner-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 anautomorphism.
- 4.
Moreover, relation
(3) can be taken as the definitionof a unitary transformation. Indeed, using the polarizationidentity
it is possible to show that if preserves the norm, then (1) must hold as well.
- 5.
A simple example of a unitary matrix is the change ofcoordinates matrix between two orthonormal bases. Indeed, let and be two orthonormal bases, andlet be the corresponding change of basis matrixdefined by
Substituting the above relation into the defining relations for anorthonormal basis,
we obtain
In matrix notation, the above is simply
as desired.
- 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.
- 7.
Unitary spaces, transformations
, matrices and operators are of fundamentalimportance 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 |