Unitary Transformation

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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 ?
For Phi-transform based communication system the unitary transform matrix is defined as follows
This third edition adds coverage of transforms including finite Hankel transforms, Legendre transforms, Jacobi and Gengenbauer transforms, fraction Fourier transforms, Zak transforms, multidimensional discrete unitary transforms, and Hilbert-Huang transforms.