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Encyclopedia > Orthogonality

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. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Perpendicular is a geometric term that may be used as a noun or adjective. ... This article is about angles in geometry. ...


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. In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...


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. In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i. ...


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, In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

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. Rewriting in mathematics, computer science and logic covers a wide range of methods of transforming strings, written in some fixed alphabet, that are not deterministic but are governed by explicit rules. ... Orthogonality as a property of term rewriting systems describes where the reduction rules of the system are all left-linear, that is each variable occurs only once on the left hand side of each reduction rule, and there is no overlap between them. ... Confluence is a property of term rewriting systems, describing that terms in this system can be rewritten in more than one way, to yield the same result. ...


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. In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ...

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. 2-dimensional renderings (ie. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ... The radian is a unit of plane angle. ... Perpendicular is a geometric term that may be used as a noun or adjective. ...


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


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. In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ... In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...


SOMEONE PLEASE PROVE WHY Non-zero pairwise orthogonal vectors are always LINEARLY INDEPENDENT and state its significance (I know this is a very important premise for other things).


Orthogonal functions

It is common to use the following inner product for two functions f and g: Partial plot of a function f. ...

langle f, grangle_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. A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a weight than others. ...


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 In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

||f||_w = sqrt{langle f, frangle_w}

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

  • orthogonal if
langle f_i, f_j rangle=int_{-infty}^infty f_i(x) f_j(x) w(x),dx=||f_i||^2delta_{i,j}=||f_j||^2delta_{i,j}
  • orthonormal
langle f_i, f_j rangle=int_{-infty}^infty f_i(x) f_j(x) w(x),dx=delta_{i,j}

where

delta_{i,j}=left{begin{matrix}1 & mathrm{if} i=j  0 & mathrm{if} ineq jend{matrix}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. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p0(x), p1(x), p2(x) ... , in which each pn(x) has degree n, and such that any two different polynomials in the sequence are orthogonal to each other in the following sense: One can define an inner...


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:
mathbf{v}_k = sum_{begin{matrix}i=0ai+k < nend{matrix}}^{n/a} mathbf{e}_i
for some positive integer a, and for 1 ≤ ka − 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,
int_{-1}^{1} left(10t^3+17t^2-{7over 9}t-{17over 3}right),dt = left[{5over 2}t^4+{17over 3}t^3-{7over 18}t^2-{17over 3}tright]_{-1}^{1}
=left({5over 2}(1)^4+{17over 3}(1)^3-{7over 18}(1)^2-{17over 3}(1)right)-left({5over 2}(-1)^4+{17over 3}(-1)^3-{7over 18}(-1)^2-{17over 3}(-1)right)
={19over 9}-{19over 9}=0.
  • 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.

In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... Fourier series are a mathematical technique for analyzing an arbitrary periodic function by decomposing the function into a sum of much simpler sinusoidal component functions, which differ from each other only in amplitude and frequency. ... In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p0(x), p1(x), p2(x) ... , in which each pn(x) has degree n, and such that any two different polynomials in the sequence are orthogonal to each other in the following sense: One can define an inner... In mathematics, the Hermite polynomials, named in honor of Charles Hermite (Hermite is pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... In mathematics, Legendre functions are solutions to Legendres differential equation: They are named after Adrien-Marie Legendre. ... In mathematics, the uniform distributions are simple probability distributions. ... In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by These polynomials are orthogonal to each other with respect to the inner product given by Also, for each n, Ln(x) is a solution of Laguerres equation which is a second-order... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics, the gamma distribution is a continuous probability distribution. ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivres formula and which are easily defined recursively, like Fibonacci or Lucas numbers. ... The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [−R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse...

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'. A square in two-point perspective. ... A vanishing point is a point in a perspective drawing where parallel lines seem to converge into. ...


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. Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... An instruction set, or instruction set architecture (ISA), describes the aspects of a computer architecture visible to a programmer, including the native datatypes, instructions, registers, addressing modes, memory architecture, interrupt and exception handling, and external I/O (if any). ... In computer architecture, a processor register is a small amount of very fast computer memory used to speed the execution of computer programs by providing quick access to commonly used values—typically, the values being in the midst of a calculation at a given point in time. ... Addressing modes, a concept from computer science, are an aspect of the instruction set architecture in most central processing unit (CPU) designs. ...


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. In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...


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. General Information Generically (as a multiplexing scheme), code division multiple access (CDMA) is any use of any form of spread spectrum by multiple transmitters to send to the same receiver on the same frequency channel at the same time without harmful interference. ... Time Division Multiple Access (TDMA) is a technology for shared medium (usually radio) networks. ... FDMA, or frequency-division multiple access, is the oldest and most important of the three main ways for multiple radio transmitters to share the radio spectrum. ...


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. Synergy or synergism (from the Greek synergos meaning working together, circa 1660) refers to the phenomenon in which two or more discrete influences or agents acting together create an effect greater than the sum of the effects each is able to create independently. ...


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. Taxonomy (from Greek verb tassein = to classify and nomos = law, science, cf economy) may refer to: the science of classifying living things (see alpha taxonomy) a system of classification in some other field Taxonomy was once only the science of classifying living organisms, but later the word was applied in...


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. A Latin square is an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column. ...


Quantum mechanics

In quantum mechanics, two eigenstates of a wavefunction, ψm and ψn, are orthogonal unless they are identical (i.e. m=n). This means, in Dirac notation, that < ψm | ψn > = 0 unless m=n, in which case < ψm | ψn > = 1. The fact that < ψm | ψn > = 1 is because wavefunctions are normalized. For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... In quantum mechanics, operators correspond to observable variables, eigenvectors are also called eigenstates, and the eigenvalues of an operator represent those values of the corresponding variable that have non-zero probability of occurring. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... Broadly, normalization (also spelled normalisation) is any process that makes something more normal, which typically means conforming to some regularity or rule, or returning from some state of abnormality. ...


See also

In linear algebra, orthogonalization means the following: we start with vectors v1,...,vk in an inner product space, most commonly the Euclidean space Rn which are linearly independent and we want to find mutually orthogonal vectors u1,...,uk which generate the same subspace as the vectors v1,...,vk. ... In mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ... In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i. ... In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ... In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. ... In mathematics, an orthonormal basis of an inner product space V(i. ... In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p0(x), p1(x), p2(x) ... , in which each pn(x) has degree n, and such that any two different polynomials in the sequence are orthogonal to each other in the following sense: One can define an inner... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

References and external links


  Results from FactBites:
 
Orthogonality - Wikipedia, the free encyclopedia (1338 words)
Hence orthogonality of vectors is a generalization of the concept of perpendicular.
In 4D the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.
The Legendre polynomials are orthogonal with respect to the uniform distribution on the interval from −1 to 1.
Orthogonal group - Wikipedia, the free encyclopedia (1448 words)
In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.
For the usual orthogonal group over the reals it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.
As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant.
  More results at FactBites »

 
 

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