Unitary transformation

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In mathematics, a unitary transformation may be informally defined as 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.

Formal definition[edit]

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\to H_2\,

where H_1 and H_2 are Hilbert spaces, such that

\langle Ux, Uy \rangle = \langle x, y \rangle

for all x and y in H_1.

Properties[edit]

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

Unitary operator[edit]

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[edit]

A closely related notion is that of antiunitary transformation, which is a bijective function

U:H_1\to H_2\,

between two complex Hilbert spaces such that

\langle Ux, Uy \rangle = \overline{\langle x, y \rangle}=\langle y, x \rangle

for all x and y in H_1, where the horizontal bar represents the complex conjugate.

See also[edit]