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Orthogonality
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In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek (orthos), meaning "straight", and (gonia), meaning "angle". For example, a subway and the street above, although they do not physically intersect, are orthogonal if they cross at a right angle.
Definitions
Definition. Two vectors and in an inner product space are orthogonal if their inner product is zero.
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Encyclopedia
In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek (orthos), meaning "straight", and (gonia), meaning "angle". For example, a subway and the street above, although they do not physically intersect, are orthogonal if they cross at a right angle.
Definitions
Definition. Two vectors and in an inner product space are orthogonal if their inner product is zero. This situation is denoted .
Definition. Two vector subspaces and of an inner product space are called orthogonal subspaces if each vector in is orthogonal to each vector in . The largest subspace that is orthogonal to a given subspace is its orthogonal complement.
Definition. A linear transformation is called an orthogonal linear transformation if it preserves the inner product. That is, for all pairs of vectors and in the inner product space ,
This means that preserves the angle between and ,
and that the lengths of and 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.
Several vectors are called pairwise orthogonal if any two of them are orthogonal, and a set of such vectors is called an orthogonal set. Non-zero pairwise orthogonal vectors are always linearly independent.
Note that the word normal is sometimes also used in place of orthogonal. However, normal can also refer to unit vectors. In particular, an orthogonal set is called orthonormal if all its vectors are unit vectors. 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 p/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.
Orthogonal functions
It is common to use the following inner product for two functions f and g:
Here we introduce a nonnegative weight function in the definition of this inner product.
We say that those functions are orthogonal if that inner product is zero:
We write the norms with respect to this inner product and the weight function as
The members of a sequence are:
- orthogonal if
- orthonormal
where
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:
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- 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,
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- The functions 1, sin(nx), cos(nx) : n = 1, 2, 3, ... are orthogonal with respect to Lebesgue measure on the interval from 0 to 2p. This fact is basic in the theory of Fourier series.
- In quantum mechanics, two eigenstates of a wavefunction, and , are orthogonal if they correspond to different eigenvalues. This means, in Dirac notation, that unless and correspond to the same eigenvalue. This follows from the fact 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 and architecture
In art the perspective imagined lines pointing to the vanishing point are referred to as 'orthogonal lines'.
The term "orthogonal line" often has a quite different meaning in the literature of modern art criticism. Many works by painters such as Piet Mondrian and Burgoyne Diller are noted for their exclusive use of "orthogonal lines" — not, however, with reference to perspective, but rather referring to lines which are straight and exclusively horizontal or vertical, forming right angles where they intersect. For example, an essay at the website of the Thyssen-Bornemisza Museum states that "Mondrian ....dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours."
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
External links
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