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 standard example of a unitary space is with inner product
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.
and hence by the inner-product axioms that
Thus, the kernel of is trivial, and therefore it is an automorphism.
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,
In matrix notation, the above is simply
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.
|Date of creation||2013-03-22 12:02:01|
|Last modified on||2013-03-22 12:02:01|
|Last modified by||asteroid (17536)|
|Synonym||complex inner product space|