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Unitary transformation

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Title: Unitary transformation  
Author: World Heritage Encyclopedia
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Subject: Localized molecular orbitals, Coherent control, Wigner's theorem, Schrieffer–Wolff transformation, Bogoliubov transformation
Collection: Functional Analysis, Linear Algebra
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Unitary transformation

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 1
  • Properties 2
  • Unitary operator 3
  • Antiunitary transformation 4
  • See also 5

Formal definition

More precisely, a unitary transformation is an isomorphism between two Hilbert spaces. In other words, a unitary transformation is a bijective function

U:H_1\to H_2\,

where H_1 and H_2 are Hilbert spaces, such that

\langle Ux, Uy \rangle_{H_2} = \langle x, y \rangle_{H_1}

for all x and y in H_1.

Properties

A unitary transformation is an isometry, as one can see by setting x=y in this formula.

Unitary operator

In the case when H_1 and H_2 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

U:H_1\to H_2\,

between two complex Hilbert spaces such that

\langle Ux, Uy \rangle = \overline{\langle x, y \rangle}=\langle y, x \rangle

for all x and y in H_1, where the horizontal bar represents the complex conjugate.

See also

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