Unitary Transformation
Get Unitary Transformation essential facts below. View Videos or join the Unitary Transformation discussion. Add Unitary Transformation to your PopFlock.com topic list for future reference or share this resource on social media.
Unitary Transformation

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

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

where ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ are Hilbert spaces, such that

${\displaystyle \langle Ux,Uy\rangle _{H_{2}}=\langle x,y\rangle _{H_{1}}}$

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

Properties

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

Unitary operator

In the case when ${\displaystyle H_{1}}$ and ${\displaystyle H_{2}}$ are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.

Antiunitary transformation

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

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

between two complex Hilbert spaces such that

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

for all ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle H_{1}}$, where the horizontal bar represents the complex conjugate.