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 ?
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.
In the real case, the unitary transformations can be chosen to be real orthogonal, but to simplify, in the following when speaking of unitary transformations, we implicitly mean real orthogonal transformations in the case of real matrices.
The second unitary transformation is an inversion about average operation.
Calculations are carried out on vectors by quantum gates that apply unitary transformations to these vectors to produce other vectors.
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.
Different fluids have different unitary transformations and so carry out different calculations.
Although most of the unitary transformations can only be implemented by approximation very inefficiently, that is the number of fault-tolerant gates is exponential in the number of qubits of the operator, it may be possible that some universal bases are more efficient than others to approximate some specific set of unitary operators.
Focusing mainly on the unitary transformations, which the Fourier transform shares with other transforms that have been derived from it, they seek to help graduate students and engineers use new forms and methods of signals and images in the frequency domain, as well as in the so-called frequency-and-time domain.
Background appendices are included on mathematical concepts, quantum measurement, the harmonic oscillator, and unitary transformations.
For the transformation from the Cartesian OXYZ into the O'X'Y'Z' the following unitary transformations have been used:
Furthermore, all the computations are carried out using unitary transformations and, therefore, the algorithms are both fast and numerically robust.
n] into tridiagonal form by unitary transformations.