Orthogonality


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orthogonality

[ȯr‚thäg·ə′nal·əd·ē]
(mathematics)
Two geometric objects have this property if they are perpendicular.
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 It might be outdated or ideologically biased.

Orthogonality

 

a generalization and often a synonym of the concept of perpendicularity. If two vectors in three-dimensional space are perpendicular, their scalar product is equal to zero. This fact permits us to generalize the concept of perpendicularity by extending it to vectors in any linear space, where a scalar product is defined with the usual properties. Thus two vectors are said to be orthogonal if their scalar product is equal to zero. In particular, let us define the scalar product in the space of complex-valued functions on the interval [a, b] by the formula

where ρ(x) ≥ 0. Then, if (f, ϕ)ρ = 0, that is,

f(x) and ϕ(x) are said to be orthogonal with respect to the weight function ρ(x). Two linear subspaces are termed orthogonal if every vector in one of them is orthogonal to every vector in the other. This concept generalizes the concept of the perpendicularity of two lines or of a line and a plane in three-dimensional space but not the concept of the perpendicularity of two planes. Curves that intersect at right angles, as measured by the angle between the tangents at the point of intersection, are called orthogonal curves.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
SOROKIN, Rational Approximations and Orthogonality, Translations of Mathematical Monographs 92, Amer.
As it is stated in the perturbation theory, eigenfunctions are able to form a basis that allows to span the space of solutions, if completeness and orthogonality [mathematical expression not reproducible] are satisfied [4], which are the main difficulties in applying the perturbation theory to (6).
Away from the aligned path 1, SNRs decrease to 0 dB which means degradation of orthogonality. Even along the path 1, the orthogonality is broken over the distance of 1.35 due to the reflection on the wall, floor, and ceiling.
Next, for each i [greater than or equal to] 1 and j = 0, ..., i, multiplying both sides of (33) by [C.sup.j.sub.i]([eta], [phi]), integrating it over S, and using the orthogonality relation (13a), (13b), and (13c), we obtain
Before proceeding, we mention an important caveat with regards to the orthogonality thesis, namely, that it is not a free orthogonality.
This way, the modulation system now adopts a block-based signal structure, so that a CP can be added to provide orthogonality against multipath channels without affecting signal's TFL.
For a generalization of this notion of orthogonality to the case of orthogonal sequences in complex Banach spaces, we refer the reader to [8].
Our construction of D-gD system is largely owing to the orthogonality between [[PSI].sub.F] and [[PSI].sub.H].
There are four main sources of errors in a turntable system; they are position error, wobble error, orthogonality error, and intersection error.
In this paper, using the near orthogonality property, we provide a recovery condition for GBCD under the total perturbations.
To apply the MAM to a forced vibration analysis of a Timoshenko beam, the orthogonality properties of mode shapes are essential.

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