Orthogonality

Overview

In mathematics, orthogonal, as a simple adjective not part of a longer phrase, is a generalization of perpendicular. It means "at right angles". The word comes from the Greek Template:Polytonic (orthos), meaning "straight", and Template:Polytonic (gonia), meaning "angle". Two streets that cross each other at a right angle are orthogonal to one another. In recent years, "perpendicular" has come to be used more in relation to right angles outside of a coordinate plane context, whereas "orthogonal" is used when discussing vectors or coordinate geometry.

Explanation

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 vector subspaces <math>A</math> and <math>B</math> of vector space <math>V</math> are called orthogonal subspaces if each vector in <math>A</math> is orthogonal to each vector in <math>B</math>. 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 unit vectors. In particular, orthonormal refers to a collection of vectors that are both orthogonal and normal (of unit length). So, using the term normal to mean "orthogonal" is often avoided.

In Euclidean vector spaces

In 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. In terms of Euclidean subspaces, the orthogonal complement of a line is the plane perpendicular to it, and vice versa. Note however that there is no correspondence with regards to perpendicular planes, because vectors in subspaces start from the origin.

In 4-dimensional Euclidean space, 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. Such a set is an orthonormal set if all its vectors are 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:

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

Here we introduce a nonnegative weight function <math>w(x)</math> in the definition of this inner product.

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

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

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

<math>||f||_w = \sqrt{\langle f, f\rangle_w}</math>

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 the Kronecker delta. In other words, any two of them are orthogonal, and the norm of each is 1 in the case of the orthonormal sequence. See in particular orthogonal polynomials.

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>

<math>={19\over 9}-{19\over 9}=0.</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.

• In quantum mechanics, two eigenstates of a wavefunction, <math> \psi_m </math> and <math> \psi_n </math>, are orthogonal if they correspond to different eigenvalues. This means, in Dirac notation, that <math> \langle \psi_m | \psi_n \rangle = 0 </math> unless <math> \psi_m </math> and <math> \psi_n </math> correspond to the same eigenvalue. This follows from that Schrödinger's equation is a Sturm-Liouville equation (in Schrödinger's formulation) or that observables are given by hermitian operators (in Heisenberg's formulation).

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

Orthogonality is a system design property facilitating feasibility and compactness of complex designs. 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 behavior 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 is easier to verify designs that neither cause side effects nor depend on them.

For example, a car has orthogonal components and controls (e.g. accelerating the vehicle does not influence anything else but the components involved exclusively with the acceleration function). On the other hand, a non-orthogonal design might have its steering influence its braking (e.g. Electronic Stability Control), or its speed tweak its suspension. 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.

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.

Radio communications

In radio communications, multiple-access schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals using different basis functions than the desired signal. One such scheme is TDMA, where the orthogonal basis functions are non-overlapping rectangular pulses ("time slots").

Another scheme is orthogonal frequency-division multiplexing (OFDM), which refers to the use, by a single transmitter, of a set of frequency multiplexed signals with the exact minimum frequency spacing needed to make them orthogonal so that they do not interfere with each other. Well known examples include (a and g) versions of 802.11 Wi-Fi; Wimax; DVB-T, the terrestrial digital TV broadcast system used in most of the world outside North America; and DMT, the standard form of ADSL.

Statistics, econometrics, and economics

When performing statistical analysis, variables that affect a particular result are said to be orthogonal if they are uncorrelated. That is to say that by varying each separately, one can predict the combined effect of varying them jointly. If correlation is present, the factors are not orthogonal. In addition, orthogonality restrictions are necessary for inference. This meaning of orthogonality 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.

Chemistry

In chemistry orthogonal protection is a strategy allowing the deprotection of functional groups independently of each other.

See also

• orthogonalization
  • Gram-Schmidt process
• orthogonal complement
• orthonormality
• Pan-orthogonality occurs in coquaternions
• orthonormal basis
• orthogonal polynomials
• orthogonal matrix
• orthogonal group
• surface normal

External links

• Chapter 4 - Compactness and Orthogonality in The Art of Unix Programming