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Orthogonality

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In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. It means at right angles. It comes from the Greek orthos, meaning "straight", used by Euclid to mean right; and gonia, meaning angle. Two streets that cross each other at a right angle are orthogonal to one another.

Formally, two vectors [x] and [y] in an inner product space [V] are orthogonal if their inner product [\langle x, y \rangle] is zero. This situation is denoted [x \perp y].

Two subspaces [A] and [B] of [V] are called orthogonal subspaces if each vector in [A] is orthogonal to each vector in [B]. Note however that this does not correspond with the geometric concept of perpendicular planes. The largest subspace that is orthogonal to a given subspace is its orthogonal complement.

A linear transformation [T : V \rightarrow V] is called an orthogonal linear transformation if it preserves the inner product. That is, for all pairs of vectors [x] and [y] in the inner product space [V],

[\langle Tx, Ty \rangle = \langle x, y \rangle.]
This means that [T] preserves the angle between [x] and [y], and that the lengths of [Tx] and [x] are equal.

A term rewriting system is said to be orthogonal if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent.

The word normal is sometimes also used in place of orthogonal. However, normal can also refer to vectors of unit length. In particular, orthonormal refers to a collection of vectors that are both orthogonal and of unit length. So the orthogonal usage of the term normal is often avoided.

Contents

In Euclidean vector spaces

For example, in a 2- or 3-dimensional Euclidean space, two vectors are orthogonal if their dot product is zero, i.e., they make an angle of 90° or π/2 radians. Hence orthogonality of vectors is a generalization of the concept of perpendicular. Also a line through the origin is orthogonal to a plane through the origin if they are perpendicular. Note however that there is no correspondence with regard to perpendicular planes.

In 4D the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.

Several vectors are called pairwise orthogonal if any two of them are orthogonal, and a set of such vectors is called an orthogonal set. They are said to be orthonormal if they are all unit vectors. Non-zero pairwise orthogonal vectors are always linearly independent.

Orthogonal functions

It is common to use the following inner product for two functions f and g:

[\langle f, g\rangle_w = \int_a^b f(x)g(x)w(x)\,dx.]
Here we introduce a nonnegative weight function [w(x)] in the definition of this inner product.

We say that those functions are orthogonal if that inner product is zero:

[\int_a^b f(x)g(x)w(x)\,dx = 0.]
We write the norms with respect to this inner product and the weight function as
[||f||_w = \sqrt]
The members of a sequence are:
[\langle f_i, f_j \rangle=\int_^\infty f_i(x) f_j(x) w(x)\,dx=||f_i||^2\delta_=||f_j||^2\delta_]
  • orthonormal
    • [\langle f_i, f_j \rangle=\int_^\infty f_i(x) f_j(x) w(x)\,dx=\delta_]
    where
    [\delta_=\left\1 & \mathrm\ i=j \\ 0 & \mathrm\ i\neq j\end\right\}]
    is Kronecker's delta. In other words, any two of them are orthogonal and the norm of each is 1. See in particular orthogonal polynomials.

    Examples

    :[\mathbf_k = \sum_i=0\\ai+k < n\end}^ \mathbf_i]
    for some positive integer a, and for 1 ≤ k ≤ a − 1, these vectors are orthogonal, for example (1, 0, 0, 1, 0, 0, 1, 0)T, (0, 1, 0, 0, 1, 0, 0, 1)T, (0, 0, 1, 0, 0, 1, 0, 0)T are orthogonal.
    :[\int_^ \left(10t^3+17t^2-t-\right)\,dt = \left[t^4+t^3-t^2-tright]_^]
    :[=\left((1)^4+(1)^3-(1)^2-(1)\right)-\left((-1)^4+(-1)^3-(-1)^2-(-1)\right)]
    :[=-=0.]

    Derived meanings

    Other meanings of the word orthogonal evolved from its earlier use in mathematics.

    Art

    In art the perspective imagined lines pointing to the vanishing point are referred to as 'orthogonal lines'.

    Computer science

    In computer science, an instruction set is said to be orthogonal if any instruction can use any register in any addressing mode. This terminology results from considering an instruction as a vector whose components are the instruction fields. One field identifies the registers to be operated upon, and another specifies the addressing mode. An orthogonal instruction set uniquely encodes all combinations of registers and addressing modes.

    Orthogonality is a system design property which enables the making of complex designs feasible and compact. The aim of an orthogonal design is to guarantee that operations within one of its components neither create nor propagate side-effects to other components. For example a car has orthogonal components and controls, e.g. accelerating the vehicle does not influence anything else but the components involved in the acceleration. On the other hand, a car with non-orthogonal design might have, for example, the acceleration influencing the radio tuning or the display of time. Consequently, this usage is seen to be derived from the use of orthogonal in mathematics; one may project a vector onto a subspace by projecting it onto each member of a set of basis vectors separately and adding the projections if and only if the basis vectors are mutually orthogonal.

    Orthogonality guarantees that modifying the technical effect produced by a component of a system neither creates nor propagates side effects to other components of the system. The emergent behaviour of a system consisting of components should be controlled strictly by formal definitions of its logic and not by side effects resulting from poor integration, i.e. non-orthogonal design of modules and interfaces. Orthogonality reduces testing and development time, because it's easier to verify designs that neither cause side effects nor depend on them.

    Radio communications

    In radio communications, multiple access schemes are orthogonal when a receiver can (theoretically) completely reject an arbitrarily strong unwanted signal. An example of an orthogonal scheme is Code Division Multiple Access, CDMA. Examples of non-orthogonal schemes are TDMA and FDMA.

    Social sciences/Statistics/Econometrics

    In the social sciences, variables that affect a particular result are said to be orthogonal if they are independent. That is to say that by varying each separately, one can predict the combined effect of varying them jointly. If synergistic effects are present, the factors are not orthogonal. This meaning derives from the mathematical one, because orthogonal vectors are linearly independent.

    Taxonomy

    in taxonomy, an orthogonal classification is one in which no item is a member of more than one group, that is, the classifications are mutually exclusive.

    Combinatorics

    In combinatorics, two n×n Latin squares are said to be orthogonal if their superimposition yields all possible n2 combinations of entries. One can also have a [more general definition] of combinatorial orthogonality.

    Quantum mechanics

    In quantum mechanics, two eigenstates of a wavefunction, [ \psi_m ] and [ \psi_n ], are orthogonal unless they are identical (i.e. m=n). This means, in Dirac notation, that [ < \psi_m | \psi_n > = 0 ] unless m=n, in which case [ < \psi_m | \psi_n > = 1 ]. The fact that [ < \psi_m | \psi_n > = 1 ] is because wavefunctions are normalized.

    See also

    References and external links

     

    From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
    All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.

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