# Unitary Transformation

(redirected from Antiunitary)

## unitary transformation

[′yü·nə‚ter·ē ‚tranz·fər′mā·shən]
(mathematics)
A linear transformation on a vector space which preserves inner products and norms; alternatively, a linear operator whose adjoint is equal to its inverse.

## Unitary Transformation

a linear transformation

with complex coefficients that leaves invariant the sum of the squares of the absolute values of the transformed quantities

Since it preserves the length

of a vector x with components x1,x2, . . ., xn, a unitary transformation is the extension to a complex n-dimensional vector space of the notion of a rotation in the Euclidean plane or in Euclidean 3-space. The coefficients of a unitary transformation form a unitary matrix. The unitary transformations of an n-dimensional complex space form a group under multiplication. If the coefficients uij and the transformed quantities xi are real, then the unitary transformation reduces to an orthogonal transformation of an n-dimensional real vector space.

References in periodicals archive ?
We note also that if the Hilbert space underlying the Hilbertian proximity space of such a quantization is separable and infinite dimensional, then it follows from a celebrated theorem of Wigner that [Sigma] must be the restriction of a unitary or an antiunitary transformation in the original Hilbert space [Varadarajan 1984, p.

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