Unitary transformation

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

${\displaystyle U:H_{1}\to H_{2}}$

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

${\displaystyle \langle Ux,Uy\rangle _{H_{2}}=\langle x,y\rangle _{H_{1}}\quad {\text{ for all }}x,y\in H_{1}.}$

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

In the case when ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.

In complex coordinate space ${\displaystyle \mathbb {C} ^{n}}$ unitary transformations always have the shape

${\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} ^{n},\,x\mapsto U\cdot x}$,

where ${\displaystyle U\in \mathbb {C} ^{n\times n}}$ is a unitary matrix, and the dot before the vector is the matrix-vector-product.

This matrix ${\textstyle U}$ satisfies ${\textstyle UU^{*}=I}$ ^[2].

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

${\displaystyle U:H_{1}\to H_{2}\,}$

between two complex Hilbert spaces such that

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

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

• Antiunitary
• Orthogonal transformation
• Time reversal
• Unitary group
• Unitary operator
• Unitary matrix
• Wigner's theorem
• Unitary transformations in quantum mechanics