Unitary transformation
In mathematics, a unitary transformation is a linear isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, a unitary transformation is an isometric isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a unitary transformation is a bijective function
- [math]\displaystyle{ U : H_1 \to H_2 }[/math]
between two inner product spaces, [math]\displaystyle{ H_1 }[/math] and [math]\displaystyle{ H_2, }[/math] such that
- [math]\displaystyle{ \langle Ux, Uy \rangle_{H_2} = \langle x, y \rangle_{H_1} \quad \text{ for all } x, y \in H_1. }[/math]
It is a linear isometry, as one can see by setting [math]\displaystyle{ x=y. }[/math]
Unitary operator
In the case when [math]\displaystyle{ H_1 }[/math] and [math]\displaystyle{ H_2 }[/math] 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
- [math]\displaystyle{ U:H_1\to H_2\, }[/math]
between two complex Hilbert spaces such that
- [math]\displaystyle{ \langle Ux, Uy \rangle = \overline{\langle x, y \rangle}=\langle y, x \rangle }[/math]
for all [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] in [math]\displaystyle{ H_1 }[/math], where the horizontal bar represents the complex conjugate.
See also
- Antiunitary
- Orthogonal transformation
- Time reversal
- Unitary group
- Unitary operator
- Unitary matrix
- Wigner's theorem
- Unitary transformations in quantum mechanics
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