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Endomorphism preserving the inner product
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
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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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e0f258e7f79c10808827282956dec6360f67e9)
between two inner product spaces,
and
such that
![{\displaystyle \langle Ux,Uy\rangle _{H_{2}}=\langle x,y\rangle _{H_{1}}\quad {\text{ for all }}x,y\in H_{1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9215728b1b626c3e122b41a434da349656317a8e)
It is a
linear isometry
, as one can see by setting
Unitary operator
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In the case when
and
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
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A closely related notion is that of
antiunitary
transformation
, which is a bijective function
![{\displaystyle U:H_{1}\to H_{2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8dc0d948f12b655141ed2d97b3eb75c8b850113)
between two
complex
Hilbert spaces such that
![{\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}=\langle y,x\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d25b5ba7ccd5886de1c8f89650df3f38044c78c)
for all
and
in
, where the horizontal bar represents the
complex conjugate
.
See also
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