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Unitary transformation

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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 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 H 2

where H 1 and H 2 are Hilbert spaces, such that

U x , U y H 2 = x , y 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 H 2

between two complex Hilbert spaces such that

U x , U y = x , y ¯ = y , x

for all x and y in H 1 , where the horizontal bar represents the complex conjugate.

References

Unitary transformation Wikipedia


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