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.

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 $H_{1}$ . A unitary transformation is an isometry, as one can see by setting $x=y$ in this formula.

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.

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.

Unitary transformation

See also