Unitary transformation
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In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Contents
Formal definition
More precisely, a unitary transformation is an isomorphism between two Hilbert spaces. In other words, a unitary transformation is a bijective function
where and are Hilbert spaces, such that
for all and in .
Properties
A unitary transformation is an isometry, as one can see by setting in this formula.
Unitary operator
In the case when and are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.
Antiunitary transformation
A closely related notion is that of antiunitary transformation, which is a bijective function
between two complex Hilbert spaces such that
for all and in , where the horizontal bar represents the complex conjugate.
See also
- Antiunitary
- Orthogonal transformation
- Time reversal
- Unitary group
- Unitary operator
- Unitary matrix
- Wigner's Theorem
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