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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 definitionEdit

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  .

PropertiesEdit

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

Unitary operatorEdit

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 transformationEdit

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 alsoEdit