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 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}}
between two inner product spaces, H 1 {\displaystyle H_{1}} and 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}.}
It is a linear isometry, as one can see by setting x = y . {\displaystyle x=y.}
Unitary operator In the case when H 1 {\displaystyle H_{1}} and 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.
Relation to unitary matrices In complex coordinate space C n {\displaystyle \mathbb {C} ^{n}} unitary transformations always have the shape
f : C n → C n , x ↦ U ⋅ x {\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} ^{n},\,x\mapsto U\cdot x} , where U ∈ C n × n {\displaystyle U\in \mathbb {C} ^{n\times n}} is a unitary matrix, and the dot before the vector is the matrix-vector-product. This matrix U {\textstyle U} satisfies U U ∗ = I {\textstyle UU^{*}=I} .
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}\,}
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 }
for all x {\displaystyle x} and y {\displaystyle y} in H 1 {\displaystyle H_{1}} , where the horizontal bar represents the complex conjugate.
See also Antiunitary Orthogonal transformation Time reversal Unitary group Unitary operator Unitary matrix Wigner's theorem Unitary transformations in quantum mechanics
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