Orthogonal Transformation

orthogonal transformation

[ȯr′thäg·ən·əl ‚tranz‚fər′mā·shən]
A linear transformation between real inner product spaces which preserves the length of vectors.

Orthogonal Transformation


a linear transformation of a Euclidean vector space that preserves the lengths or (equivalently) the scalar products of vectors. In an orthonormal basis an orthogonal transformation corresponds to an orthogonal matrix. Orthogonal transformations form a group, the group of rotations of the given Euclidean space about the origin. In three-dimensional space an orthogonal transformation reduces to a rotation through a certain angle about some axis passing through the origin O, if the determinant of the corresponding orthogonal matrix is +1. If the determinant is —1, then the rotation must be supplemented by a reflection in the plane passing through O perpendicular to the axis of rotation. In two-dimensional space, that is, in a plane, an orthogonal transformation defines a rotation through a certain angle about O or a reflection relative to some line passing through O. Orthogonal transformations are used to reduce a quadratic form to the principal axes.

References in periodicals archive ?
The triangularization process can be realized by any of the orthogonal transformation components; Givens rotations, Householder transformation and Modified Gram-Schmidt orthogonalization [6,8].
Since the vector norm is invariant of the orthogonal transformation Q(k)[1], we apply QR decomposition to transform the information matrix to generate an upper right triangular matrixR(k).
We define the orthogonal transformation matrix Q(n) as in (2.
To complete the orthogonal reduction, we apply an orthogonal transformation that involves the bottom p +1 rows of the system and produces a system of the form
This is followed by an orthogonal transformation involving the bottom p + 1 rows of the system as before.
Principal component analysis is an optimal image compression orthogonal transformation, the aim is to find a set of data space vectors to interpret the data as much as possible the variance of the data from the R-dimensional space down to the original M-dimensional (R [much greater than] M), after the dimensionality of the data stored in the main information to make the data easier to handle.
Unlike the Fourier transform of this transformation, discrete cosine transform, orthogonal transformation, which transforms the transform kernel is fixed, while the KL transform is a collection of images with different statistical properties of the nuclear matrix have different transformations that transform the nuclear matrix is a collection of images to determine the statistical properties, so the discrete KL transform is a transformation based on the demographic characteristics of the image.
7) can be constructed efficiently as the product of 2 x 2 orthogonal transformations.
The remaining orthogonal transformations in Z are either Givens rotations or Householder transformations applied to only two rows.
OLDA obtains easily the optimal transformation matrix by only orthogonal transformations without computing any eigen-decomposition and matrix inverse.
He assumes readers to be familiar with basic techniques of numerical linear algebra such as norm estimates, orthogonal transformations, and factorizations.
3 guarantees that corresponding condensed forms under orthogonal transformations also exist, see also [29].

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