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 <math>x</math> and <math>y</math> in an inner product space <math>V</math> are orthogonal if their inner product <math>\langle x, y \rangle</math> is zero. This situation is denoted <math>x \perp y</math>.

Two subspaces <math>A</math> and <math>B</math> of <math>V</math> are called orthogonal subspaces if each vector in <math>A</math> is orthogonal to each vector in <math>B</math>. 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 <math>T : V \rightarrow V</math> is called an orthogonal linear transformation if it preserves the inner product. That is, for all pairs of vectors <math>x</math> and <math>y</math> in the inner product space <math>V</math>,

<math>\langle Tx, Ty \rangle = \langle x, y \rangle.</math>

This means that <math>T</math> preserves the angle between <math>x</math> and <math>y</math>, and that the lengths of <math>Tx</math> and <math>x</math> 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.

1 In Euclidean vector spaces
2 Orthogonal functions
3 Examples
4 Derived meanings
5 See also
6 References and external links

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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.

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Orthogonal functions

We commonly use the following inner product to say that two functions f and g are orthogonal:

<math> \langle f, g \rangle = \int_a^b f(x)g(x)w(x)\,dx = 0.</math>

Here we introduce a nonnegative weight function <math>w(x)</math>, and we write

<math> \langle f, gw \rangle = \langle f, g\rangle_w</math>.

We write the norms with respect to this inner product and the weight function as

The members of a sequence { f_i : i = 1, 2, 3, ... } are:

orthogonal if

<math>\langle f_i, f_j \rangle=\int_{-\infty}^\infty f_i(x) f_j(x) w(x)\,dx=||f_i||^2\delta_{i,j}=||f_j||^2\delta_{i,j}</math>

orthonormal

<math>\langle f_i, f_j \rangle=\int_{-\infty}^\infty f_i(x) f_j(x) w(x)\,dx=\delta_{i,j}</math>

where

<math>\delta_{i,j}=\left\{\begin{matrix}1 & \mathrm{if}\ i=j \\ 0 & \mathrm{if}\ i\neq j\end{matrix}\right\}</math>

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.

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Examples

The vectors (1, 3, 2), (3, −1, 0), (1/3, 1, −5/3) are orthogonal to each other, since (1)(3) + (3)(−1) + (2)(0) = 0, (3)(1/3) + (−1)(1) + (0)(−5/3) = 0, (1)(1/3) + (3)(1) − (2)(5/3) = 0. Observe also that the dot product of the vectors with themselves are the norms of those vectors, so to check for orthogonality, we need only check the dot product with every other vector.

The vectors (1, 0, 1, 0, ...)^T and (0, 1, 0, 1, ...)^T are orthogonal to each other. Clearly the dot product of these vectors is 0. We can then make the obvious generalization to consider the vectors in Z₂ⁿ:

<math>\mathbf{v}_k = \sum_{\begin{matrix}i=0\\ai+k < n\end{matrix}}^{n/a} \mathbf{e}_i</math>

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.

Take two quadratic functions 2t + 3 and 5t² + t − 17/9. These functions are orthogonal with respect to a unit weight function on the interval from −1 to 1. The product of these two functions is 10t³ + 17t² − 7/9 t − 17/3, and now,

<math>\int_{-1}^{1} \left(10t^3+17t^2-{7\over 9}t-{17\over 3}\right)\,dt = \left[{5\over 2}t^4+{17\over 3}t^3-{7\over 18}t^2-{17\over 3}t\right]_{-1}^{1}</math>

<math>=\left({5\over 2}(1)^4+{17\over 3}(1)^3-{7\over 18}(1)^2-{17\over 3}(1)\right)-\left({5\over 2}(-1)^4+{17\over 3}(-1)^3-{7\over 18}(-1)^2-{17\over 3}(-1)\right)</math>

The functions 1, sin(nx), cos(nx) : n = 1, 2, 3, ... are orthogonal with respect to Lebesgue measure on the interval from 0 to 2π. This fact is basic in the theory of Fourier series.

Various eponymously named polynomial sequences are sequences of orthogonal polynomials. In particular:
- The Hermite polynomials are orthogonal with respect to the normal distribution with expected value 0.
- The Legendre polynomials are orthogonal with respect to the uniform distribution on the interval from −1 to 1.
- The Laguerre polynomials are orthogonal with respect to the exponential distribution. Somewhat more general Laguerre polynomial sequences are orthogonal with respect to gamma distributions.
- The Chebyshev polynomials of the first kind are orthogonal with respect to the measure <math>1/\sqrt{1-x^2}.</math>
- The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution.

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Derived meanings

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

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Art

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

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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 the test and development time, because it's easier to verify designs that neither cause side effects nor depend on them.

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Radio communications

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

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Social sciences

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 vecors are linearly independent.

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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.

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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.

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