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Unitary transformation

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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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