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

$U:H_1\to H_2\,$

where $H_1$ and $H_2$ are Hilbert spaces, such that

$\langle Ux, Uy \rangle_{H_2} = \langle x, y \rangle_{H_1}$

for all $x$ and $y$ in $H_1$.

Properties

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

Unitary operator

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.

Antiunitary transformation

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

$U:H_1\to H_2\,$

between two complex Hilbert spaces such that

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

for all $x$ and $y$ in $H_1$, where the horizontal bar represents the complex conjugate.

See also

• Antiunitary
• Orthogonal transformation
• Time reversal
• Unitary group
• Unitary operator
• Unitary matrix
• Wigner's Theorem