Orthogonality


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orthogonality

[ȯr‚thäg·ə′nal·əd·ē]
(mathematics)
Two geometric objects have this property if they are perpendicular.

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.

References in periodicals archive ?
Source: Own estimations on the basis of the data from Thomson Reuters Datastream Table 3: Orthogonality test of the forward rate forecast error Results for equation [s.
Now we introduce the following type of orthogonality.
Another discrete orthogonality relation is easily derived from the expansion of the reproducing kernel [K.
Thus, identifying restrictions that impose orthogonality of forecast errors with respect to current output impute too much information to expectations.
A New Design of Fuzzy Logic Controller Based on Generalized Orthogonality Principle, International Symposium on Computational Intelligence in Robotics and Automation, Daejon, ISBN 978-1-4244-4808-1, pp.
The derivation of the coefficients for the DFT is now easily obtained using the discrete orthogonality.
in which [perpendicular to] is an abstract orthogonality relation, was first investigated by S.
In other works, the stark orthogonality of a row of tree trunks evoked Mondrian.
Rhizomatic systems approach proposes that the focus on 'problem solving' in a real-world situation needs to shift to the process of problematization if a researcher wants to satisfy the criterion of orthogonality.
Using nonlinear GMM approach we have estimated Equation (2) with both types of orthogonality conditions we find the following results.
The orthogonality test results are consistent with rejection due to the presence of a transitory demand shock.
Due to the dispersive nature of the mobile communications channels, Multi-Carrier Code Division Multiple Access (MC-CDMA) signals will be affected by Inter Channel Interference (ICI), Inter Symbol Interference (ISI) and Multiple Access Interference (MAI), due to the loss of subcarrier orthogonality.

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