# 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.
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 It might be outdated or ideologically biased.

## 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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Section III describes stochastic processor of the orthogonal transformation. Section IV exposes theory of operation.
Only with orthogonal transformation, the original vectors can be invariant to their Euclidean distance or angle with each other.
Let [Q.sup.(j+1).sub.i] [member of] [C.sup.(j+1)Lx(j+1)L] be the orthogonal transformation which annihilates all subdiagonal entries in columns (i - 1)L +1 to iL of [[bar.H].sup.(B).sub.j] and effects no other rows so that we can write
In particular, it uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.
PCA is a statistical procedure that uses orthogonal transformation to reveal the internal structure of data, which is CME (Jackson, 1991; Williams et al., 2004).
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.
Number of techniques were developed for enforcing EBCs in the problem like Lagrange multiplier, penalty method, orthogonal transformation techniques, coupling with FEM, Nitsche's method, singular weighing functions, boundary collocation, and D'Alembert's Principle.
In order to improve the numerical instability associated with the recursive least squares(RLS) algorithm, a desirable technique based on numerically stable, robust orthogonal triangularization approach that computes the information matrix directly via QR decomposition is considered [1,3,5,6,9].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 (2.9) and (2.10) as a partition matrix
An orthogonal transformation is used to annihilate it which produces a fill-in in `b', whose annihilation produces `c'.
When uncertainties are statistically correlated, Rosenblatt transformation, Nataf transformation, and orthogonal transformation are often used to handle these uncertainties in reliability analysis.
The unstructured orthogonal transformation, however, shows the expected problems with purely imaginary eigenvalues.

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