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

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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

 

between two inner product spaces,   and   such that

 

It is a linear isometry, as one can see by setting  

Unitary operator

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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

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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

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