Unitary transformation

Informally, a unitary transformation is a transformation that respects the dot product: the dot product of two vectors before the transformation is equal to their dot 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

:$U:H\_1\; o\; H\_2,$

where $H\_1$ and $H\_2$ are Hilbert spaces, such that

:$langle\; Ux,\; Uy\; angle\; =\; langle\; x,\; y\; angle$

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

:$U:H\_1\; o\; H\_2,$

between two complex Hilbert spaces such that

:$langle\; Ux,\; Uy\; angle\; =\; overline\{langle\; x,\; y\; angle\}=langle\; y,\; x\; angle$

for all $x$ and $y$ in $H\_1$, where the horizontal bar represents the complex conjugate.

ee also

*Time reversal
*antiunitary
*Unitary group
*Unitary operator
*Unitary matrix
*Wigner's Theorem