Unitary transformation

In mathematics, a unitary transformation may be informally defined as a transformation that respects the inner product: the inner product of two vectors before the transformation is equal to their inner 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

$U:H_1\to H_2\,$

where H₁ and H₂ are Hilbert spaces, such that

$\langle Ux, Uy \rangle = \langle x, y \rangle$

for all x and y in H₁. A unitary transformation is an isometry, as one can see by setting x = y in this formula.

In the case when H₁ and H₂ 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

$U:H_1\to H_2\,$

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 H₁, where the horizontal bar represents the complex conjugate.

See also

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