Everything about Orthogonal totally explained
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 ὀρθός (
orthos), meaning "straight", and
γωνία (
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 and
in an
inner product space are orthogonal if their inner product
is zero. This situation is denoted
.
Two
vector subspaces and
of
vector space are called
orthogonal subspaces if each vector in
is orthogonal to each vector in
. The largest subspace that's orthogonal to a given subspace is its
orthogonal complement.
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's 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, for example 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's 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:
»
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 Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution.
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 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, for example 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.
For example, a car has orthogonal components and controls (for example accelerating the vehicle doesn't 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 (for example 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 don't 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're 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.
Further Information
Get more info on 'Orthogonal'.
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