- Formal definition
- Properties
- Unitary operator
- Antiunitary transformation
- See also
}}{{other uses|Transformation (mathematics) (disambiguation)}}
In mathematics, a unitary transformation is 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.
Formal definition
More precisely, a unitary transformation is an isomorphism between two Hilbert spaces. In other words, a unitary transformation is a bijective function
where 
and 
are Hilbert spaces, such that
for all 
and 
in .
Properties
A unitary transformation is an isometry, as one can see by setting 
in this formula.
Unitary operator
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
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
- Antiunitary
- Orthogonal transformation
- Time reversal
- Unitary group
- Unitary operator
- Unitary matrix
- Wigner's Theorem
- Unitary transformations in quantum mechanics
- Fatbrain
- Fatbrain.com
- Fat breakdown
- Fat butt and pancake head
- Fat Buu
- Fat Camp: An MTV Docs Movie Presentation
- Fat Cantor set
- Fatcat
- Fat cat
- Fat Cat
- FatCat
- Fat cat and friends
- Fat Cat (cartoon)
- Fat Cat (Chip 'n Dale Rescue Rangers)
- Fat cat (disambiguation)
- Fat Cat Discography
- Fat Cat (Disney)
- FatCat Records
- FATCAT RECORDS
- Fatcat Records
- Fatcat records
- Fat cat records discography
- FatCat Records discography
- Fat cats
- Fat cell