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