Unitary transformation

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.

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

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

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

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

In the case when ${\displaystyle H_{1}}${\displaystyle H_{1}} and ${\displaystyle H_{2}}${\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.

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

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

between two complex Hilbert spaces such that

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

for all ${\displaystyle x}${\displaystyle x} and ${\displaystyle y}${\displaystyle y} in ${\displaystyle H_{1}}${\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

• Linear algebra
• Functional analysis

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