# Orthogonal Transformation

## orthogonal transformation

[ȯr′thäg·ən·əl ‚tranz‚fər′mā·shən]
(mathematics)
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 ?
PCA is a statistical procedure that uses orthogonal transformation to reveal the internal structure of data, which is CME (Jackson, 1991; Williams et al.
Caption: Figure 3: A sketch of the orthogonal transformation from the physical domain (x-y coordinates) to the computational domain ([xi]-[eta] coordinates).
The unstructured orthogonal transformation, however, shows the expected problems with purely imaginary eigenvalues.
The Varimax procedure finds the orthogonal transformation of the loading matrix that maximizes the sum of those variances, summing across all m rotated factors, while the Quartimax procedure performs the same transformation across all p variables.
The triangularization process can be realized by any of the orthogonal transformation components; Givens rotations, Householder transformation and Modified Gram-Schmidt orthogonalization [6,8].
We define the orthogonal transformation matrix Q(n) as in (2.
An orthogonal transformation is used to annihilate it which produces a fill-in in `b', whose annihilation produces `c'.
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.
Optionally, if JOBU = `U', the orthogonal transformation matrix Q can also be built.
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
On output A is on upper r-Hessenberg form, B is upper triangular, and U, V are the orthogonal transformation matrices.

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