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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
The image [F.sup.(2).sub.k] of the space [F.sup.(1).sub.k] under the unitary transform [U.sub.2] = I [cross product] F is the closure of the set of all smooth functions in [L.sub.2]([R.sup.2]) which satisfy the equation
Fast Improved MUSIC Algorithm Based on Unitary Transform and MSWF
Using unitary transform and MSWF, we can get a fast improved MUSIC algorithm.
For Phi-transform based communication system the unitary transform matrix is defined as follows
Important unitary transforms have been discussed in [15].
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.