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

where H₁ and H₂ are Hilbert spaces, such that

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

between two complex Hilbert spaces such that

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

This mathematical analysis-related article is a stub. You can help Wikipedia by expanding it.