Unitary transformation
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Informally, a unitary transformation is a transformation that respects the dot product: the dot product of two vectors before the transformation is equal to their dot product after the transformation.
More precisely, a unitary transformation is an isomorphism between two Hilbert spaces. In other words, a unitary transformation is a bijective function
where H1 and H2 are Hilbert spaces, such that
for all x and y in H1. A unitary transformation is an isometry, as one can see by setting x = y in this formula.
In the case when H1 and H2 are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.
A closely related notion is that of antiunitary transformation, which is a bijective function
between two complex Hilbert spaces such that
for all x and y in H1, where the horizontal bar represents the complex conjugate.
