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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
between two inner product spaces,
and
such that
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
between two
complex
Hilbert spaces such that
for all
and
in
, where the horizontal bar represents the
complex conjugate
.
See also
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