Unitary transform

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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.[1]

Formal definition

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

U : H 1 → H 2 {\displaystyle U:H_{1}\to H_{2}} {\displaystyle U:H_{1}\to H_{2}}

between two inner product spaces, H 1 {\displaystyle H_{1}} {\displaystyle H_{1}} and H 2 , {\displaystyle H_{2},} {\displaystyle H_{2},} such that

⟨ U x , U y ⟩ H 2 = ⟨ x , y ⟩ H 1  for all  x , y ∈ H 1 . {\displaystyle \langle Ux,Uy\rangle _{H_{2}}=\langle x,y\rangle _{H_{1}}\quad {\text{ for all }}x,y\in H_{1}.} {\displaystyle \langle Ux,Uy\rangle _{H_{2}}=\langle x,y\rangle _{H_{1}}\quad {\text{ for all }}x,y\in H_{1}.}

It is a linear isometry, as one can see by setting x = y . {\displaystyle x=y.} {\displaystyle x=y.}

Unitary operator

In the case when H 1 {\displaystyle H_{1}} {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} {\displaystyle 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 {\displaystyle U:H_{1}\to H_{2}\,} {\displaystyle U:H_{1}\to H_{2}\,}

between two complex Hilbert spaces such that

⟨ U x , U y ⟩ = ⟨ x , y ⟩ ¯ = ⟨ y , x ⟩ {\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}=\langle y,x\rangle } {\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}=\langle y,x\rangle }

for all x {\displaystyle x} {\displaystyle x} and y {\displaystyle y} {\displaystyle y} in H 1 {\displaystyle H_{1}} {\displaystyle H_{1}}, where the horizontal bar represents the complex conjugate.

See also

References