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Unitary Operators / Transform

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Orthogonal transformation. In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a vector space V that has a nondegenerate symmetric bilinear form such that T preserves the bilinear form.

Orthogonal transformation

That is, for each pair u, v of elements of V, we have[1] Since the lengths of vectors and the angles between them are defined through the bilinear form, orthogonal transformations preserve lengths of vectors and angles between them. In particular, orthogonal transformations map orthonormal bases to orthonormal bases. In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix.

Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of V. Discrete Fourier transform. Relationship between the (continuous) Fourier transform and the discrete Fourier transform.

Discrete Fourier transform

Left column: A continuous function (top) and its Fourier transform (bottom). Center-left column:Periodic summation of the original function (top). Fourier transform (bottom) is zero except at discrete points. The inverse transform is a sum of sinusoids called Fourier series. Center-right column: Original function is discretized (multiplied by a Dirac comb) (top). Illustration of using Dirac comb functions and the convolution theorem to model the effects of sampling and/or periodic summation. The input samples are complex numbers (in practice, usually real numbers), and the output coefficients are complex as well. Since it deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware.

Definition[edit] The sequence of N complex numbers is transformed into an N-periodic sequence of complex numbers: , as in or . which is also N-periodic. Each . If. DFT matrix. A DFT matrix is an expression of a discrete Fourier transform (DFT) as a matrix multiplication.

DFT matrix

Definition[edit] An N-point DFT is expressed as an N-by-N matrix multiplication as , where is the original input signal, and is the DFT of the signal. The transformation of size can be defined as , or equivalently: where. Orthogonal matrix. In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. where I is the identity matrix.

Orthogonal matrix

This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: The complex analogue of an orthogonal matrix is a unitary matrix. Overview[edit] where Q is an orthogonal matrix. Examples[edit] Below are a few examples of small orthogonal matrices and possible interpretations. An instance of a 2×2 rotation matrix: Elementary constructions[edit] Lower dimensions[edit] The simplest orthogonal matrices are the 1×1 matrices [1] and [−1] which we can interpret as the identity and a reflection of the real line across the origin. Antiunitary operator. In mathematics, an antiunitary transformation, is a bijective antilinear map between two complex Hilbert spaces such that for all and in , where the horizontal bar represents the complex conjugate.

Antiunitary operator

Then U is called an antiunitary operator. Antiunitary operators are important in Quantum Theory because they are used to represent certain symmetries, such as time-reversal symmetry. Invariance transformations[edit] In Quantum mechanics, the invariance transformations of complex Hilbert space leave the absolute value of scalar product invariant: . Geometric Interpretation[edit] Unitary operator. In functional analysis, a branch of mathematics, a unitary operator (not to be confused with a unity operator) is defined as follows: The weaker condition U*U = I defines an isometry.

Unitary operator

The other condition, UU* = I, defines a coisometry. Thus a unitary operator is a bounded linear operator which is both an isometry and a coisometry.[1] An equivalent definition is the following: Definition 2. U is surjective, andU preserves the inner product of the Hilbert space, H. The following, seemingly weaker, definition is also equivalent: Definition 3. The range of U is dense in H, andU preserves the inner product of the Hilbert space, H. To see that Definitions 1 & 3 are equivalent, notice that U preserving the inner product implies U is an isometry (thus, a bounded linear operator). Examples[edit] The identity function is trivially a unitary operator.Rotations in R2 are the simplest nontrivial example of unitary operators.

Linearity[edit] Analogously you obtain. Unitary transformation. More precisely, a unitary transformation is an isomorphism between two Hilbert spaces.

Unitary transformation

In other words, a unitary transformation is a bijective function where and are Hilbert spaces, such that for all in . In this formula. In the case when are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator. A closely related notion is that of antiunitary transformation, which is a bijective function between two complex Hilbert spaces such that , where the horizontal bar represents the complex conjugate.