# 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 It might be outdated or ideologically biased.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.

References in periodicals archive ?
The method exploits infinitesimal unitary transformation steps, from which renormalization-group-like flow equations are derived to produce the effective Hamiltonian.
Following this idea, Huarng and Yeh  introduced an outstanding unitary transformation for low-complexity DOA estimation with real-valued computation.
Alhaidari [11-13] applies a unitary transformation to Dirac equation such that the resulting second-order differential equation becomes Schroodinger-like equation so that comparison with the well-known nonrelativistic problems is transparent.
The link consists of a (constant) unitary transformation of the [[beta].sup.[mu]] matrices and a projection onto a lower-dimensional subspace.
We have shown via a careful error analysis that it is possible to do this deflation via a structure preserving real orthogonal or unitary transformation as well as with a Schur complement approach.
QFT can be used in novel image encryption and decryption , where the quantum bit numbers of the quantum state |x) are m, [U.sub.QFT] is a unitary operator, and QFT is a [2.sup.m]-dimensional unitary transformation [18,19].
In this formulation the real part of the dielectric constant [[epsilon].sup./] will lead to an expression for the change in Curie temperature ([T.sub.c]) by the presence of electric field.We make use of unitary transformation discussed in previous studies  which render the most significant first order dipole moment to effect the Curie temperature ([T.sub.c]) with the applied electric field.
When anyons move around each other along braided paths, the motion changes the pattern of the anyons' interactions with each other--a change physicists call a unitary transformation. This transformation alters what will happen if two anyons collide, so by bumping anyons into each other it's possible to tell that they have moved.
Assume that U is a unitary transformation that clones, in that U(|a0>) = |aa> for all quantum states |a>.
For example, the unitary transformation that inverts the amplitude of each basis state is modeled as follows:
The nonrelativistic approximation can be obtained by using the well-known Foldy-Wouthuysen unitary transformation (FWUF)  and by neglecting the spin degree of freedom the nonrelativistic Hamiltonian is given as [25, 26]
He uses a unitary transformation which yields wave functions which do not undergo Coulomb forces.

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