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`Orthogonality
`
`In mathematics, orthogonality is the generalization of the notion
`of perpendicularity to the linear algebra of bilinear forms. Two
`elements u and v of a vector space with bilinear form B are
`orthogonal when B(u, v) = 0. Depending on the bilinear form, the
`vector space may contain nonzero self-orthogonal vectors. In the
`case of function spaces, families of orthogonal functions are used to
`form a basis.
`
`The line segments AB and CD are
`orthogonal to each other.
`
`https://en.wikipedia.org/wiki/Orthogonality
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`By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also
`has specialized meanings in other fields including art and chemistry.
`
`Contents
`Etymology
`Mathematics and physics
`Definitions
`Euclidean vector spaces
`Orthogonal functions
`Examples
`Orthogonal polynomials
`Orthogonal states in quantum mechanics
`
`Art
`Computer science
`Communications
`Statistics, econometrics, and economics
`Taxonomy
`Combinatorics
`Chemistry and biochemistry
`System reliability
`Neuroscience
`Gaming
`Other examples
`See also
`Notes
`References
`
`Etymology
`The word comes from the Greek ὀρθός (orthos), meaning "upright", and γωνία (gonia), meaning "angle". The
`ancient Greek ὀρθογώνιον orthogōnion (< ὀρθός orthos 'upright'[1] + γωνία gōnia 'angle'[2]) and classical Latin
`orthogonium originally denoted a rectangle.[3] Later, they came to mean a right triangle. In the 12th century,
`the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle.[4]
`
`Mathematics and physics
`
`Definitions
`◾ In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i.e., they form a right angle.
`
`https://en.wikipedia.org/wiki/Orthogonality
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`◾ Two vectors, x and y, in an inner product
`space, V, are orthogonal if their inner
` is zero.[6] This relationship
`product
`is denoted
`.
`◾ Two vector subspaces, A and B, of an
`inner product space V, are called
`orthogonal subspaces if each vector in
`A is orthogonal to each vector in B. The
`largest subspace of V that is orthogonal
`to a given subspace is its orthogonal
`complement.
`
`are orthogonal if their natural pairing is
`
`◾ Given a module M and its dual M∗, an
`element m′ of M∗ and an element m of M
`zero, i.e. ⟨m′, m⟩ = 0. Two sets S′ ⊆ M∗
`and S⊆ M are orthogonal if each
`
`Orthogonality and rotation of coordinate systems
`compared between left: Euclidean space through circular
`angle ϕ, right: in Minkowski spacetime through hyperbolic
`angle ϕ (red lines labelled c denote the worldlines of a
`light signal, a vector is orthogonal to itself if it lies on this
`line).[5]
`
`element of S′ is orthogonal to each
`element of S.[7]
`◾ A term rewriting system is said to be
`orthogonal if it is left-linear and is non-
`ambiguous. Orthogonal term rewriting
`systems are confluent.
`A set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is
`orthogonal. Such a set is called an orthogonal set.
`
`In certain cases, the word normal is used to mean orthogonal, particularly in the geometric sense as in the
`normal to a surface. For example, the y-axis is normal to the curve y = x2 at the origin. However, normal may
`also refer to the magnitude of a vector. In particular, a set is called orthonormal (orthogonal plus normal) if it
`is an orthogonal set of unit vectors. As a result, use of the term normal to mean "orthogonal" is often avoided.
`The word "normal" also has a different meaning in probability and statistics.
`
`A vector space with a bilinear form generalizes the case of an inner product. When the bilinear form applied to
`two vectors results in zero, then they are orthogonal. The case of a pseudo-Euclidean plane uses the term
`hyperbolic orthogonality. In the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given ϕ.
`
`Euclidean vector spaces
`In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of
`90° (π/2 radians), or one of the vectors is zero.[8] Hence orthogonality of vectors is an extension of the concept
`of perpendicular vectors to spaces of any dimension.
`
`The orthogonal complement of a subspace is the space of all vectors that are orthogonal to every vector in the
`subspace. In a three-dimensional Euclidean vector space, the orthogonal complement of a line through the
`origin is the plane through the origin perpendicular to it, and vice versa.[9]
`
`Note that the geometric concept two planes being perpendicular does not correspond to the orthogonal
`complement, since in three dimensions a pair of vectors, one from each of a pair of perpendicular planes,
`might meet at any angle.
`
`https://en.wikipedia.org/wiki/Orthogonality
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`In four-dimensional Euclidean space, the orthogonal complement of a line is a hyperplane and vice versa, and
`that of a plane is a plane.[9]
`
`Orthogonal functions
`By using integral calculus, it is common to use the following to define the inner product of two functions f and
`g with respect to a nonnegative weight function w over an interval [a, b]:
`
`In simple cases, w(x) = 1.
`
`We say that functions f and g are orthogonal if their inner product (equivalently, the value of this integral) is
`zero:
`
`Orthogonality of two functions with respect to one inner product does not imply orthogonality with respect to
`another inner product.
`
`We write the norm with respect to this inner product as
`
`The members of a set of functions {fi : i = 1, 2, 3, ...} are orthogonal with respect to w on the interval [a, b] if
`
`The members of such a set of functions are orthonormal with respect to w on the interval [a, b] if
`
`where
`
`is the Kronecker delta. In other words, every pair of them (excluding pairing of a function with itself) is
`orthogonal, and the norm of each is 1. See in particular the orthogonal polynomials.
`
`Examples
`◾ The vectors (1, 3, 2)T, (3, −1, 0)T, (1, 3, −5)T are orthogonal to each other, since (1)(3) + (3)(−1) + (2)(0) =
`0, (3)(1) + (−1)(3) + (0)(−5) = 0, and (1)(1) + (3)(3) + (2)(−5) = 0.
`
`https://en.wikipedia.org/wiki/Orthogonality
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`◾ The vectors (1, 0, 1, 0, ...)T and (0, 1, 0, 1, ...)T are orthogonal to each other. The dot product of these
`vectors is 0. We can then make the generalization to consider the vectors in Z2
`n:
`
`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.
`◾ The functions 2t + 3 and 45t2 + 9t − 17 are orthogonal with respect to a unit weight function on the interval
`from −1 to 1:
`
`◾ The functions 1, sin(nx), cos(nx) : n = 1, 2, 3, ... are orthogonal with respect to Riemann integration on the
`intervals [0, 2π], [−π, π], or any other closed interval of length 2π. This fact is a central one in Fourier
`series.
`
`Orthogonal polynomials
`◾ Various polynomial sequences named for mathematicians of the past are sequences of orthogonal
`polynomials. In particular:
`◾ The Hermite polynomials are orthogonal with respect to the Gaussian distribution with zero mean
`value.
`◾ The Legendre polynomials are orthogonal with respect to the uniform distribution on the interval
`[−1, 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
`◾ The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle
`distribution.
`
`Orthogonal states in quantum mechanics
`◾ In quantum mechanics, a sufficient (but not necessary) condition that two eigenstates of a Hermitian
`operator,
` and
`, are orthogonal is that they correspond to different eigenvalues. This means, in
`Dirac notation, that
` if
` and
` correspond to different eigenvalues. 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).
`
`Art
`In art, the perspective (imaginary) lines pointing to the vanishing point are referred to as "orthogonal lines".
`
`https://en.wikipedia.org/wiki/Orthogonality
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`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 that are straight and
`exclusively horizontal or vertical, forming right angles where they intersect. For example, an essay at the Web
`site of the Thyssen-Bornemisza Museum states that "Mondrian ... dedicated his entire oeuvre to the
`[1]
`investigation
`of
`the
`balance
`between
`orthogonal
`lines
`and
`primary
`colours."
`(http://www.museothyssen.org/thyssen_ing/coleccion/obras_ficha_texto_print497.html)
`
`Computer science
`Orthogonality in programming language design is the ability to use various language features in arbitrary
`combinations with consistent results.[10] This usage was introduced by Van Wijngaarden in the design of Algol
`68:
`
`The number of independent primitive concepts has been minimized in order that the language
`be easy to describe, to learn, and to implement. On the other hand, these concepts have been
`applied “orthogonally” in order to maximize the expressive power of the language while trying to
`avoid deleterious superfluities.[11]
`
`Orthogonality is a system design property which 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. Typically
`this is achieved through the separation of concerns and encapsulation, and it is essential for feasible and
`compact designs of complex systems. 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.
`
`An instruction set is said to be orthogonal if it lacks redundancy (i.e., there is only a single instruction that
`can be used to accomplish a given task)[12] and is designed such that instructions 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.
`
`Communications
`In communications, multiple-access schemes are orthogonal when an ideal receiver can completely reject
`arbitrarily strong unwanted signals from the desired signal using different basis functions. One such scheme is
`TDMA, where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots").
`
`https://en.wikipedia.org/wiki/Orthogonality
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`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, g, and
`n) versions of 802.11 Wi-Fi; WiMAX; ITU-T G.hn, DVB-T, the terrestrial digital TV broadcast system used in
`most of the world outside North America; and DMT (Discrete Multi Tone), the standard form of ADSL.
`
`In OFDM, the subcarrier frequencies are chosen so that the subcarriers are orthogonal to each other, meaning
`that crosstalk between the subchannels is eliminated and intercarrier guard bands are not required. This
`greatly simplifies the design of both the transmitter and the receiver. In conventional FDM, a separate filter for
`each subchannel is required.
`
`Statistics, econometrics, and economics
`When performing statistical analysis, independent variables that affect a particular dependent variable are said
`to be orthogonal if they are uncorrelated,[13] since the covariance forms an inner product. In this case the same
`results are obtained for the effect of any of the independent variables upon the dependent variable, regardless
`of whether one models the effects of the variables individually with simple regression or simultaneously with
`multiple regression. If correlation is present, the factors are not orthogonal and different results are obtained
`by the two methods. This usage arises from the fact that if centered by subtracting the expected value (the
`mean), uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data
`(i.e., vectors) and as random variables (i.e., density functions). One econometric formalism that is alternative
`to the maximum likelihood framework, the Generalized Method of Moments, relies on orthogonality
`conditions. In particular, the Ordinary Least Squares estimator may be easily derived from an orthogonality
`condition between the explanatory variables and model residuals.
`
`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.[14]
`
`Chemistry and biochemistry
`In synthetic organic chemistry orthogonal protection is a strategy allowing the deprotection of functional
`groups independently of each other. In chemistry and biochemistry, an orthogonal interaction occurs when
`there are two pairs of substances and each substance can interact with their respective partner, but does not
`interact with either substance of the other pair. For example, DNA has two orthogonal pairs: cytosine and
`guanine form a base-pair, and adenine and thymine form another base-pair, but other base-pair combinations
`are strongly disfavored. As a chemical example, tetrazine reacts with transcyclooctene and azide reacts with
`
`https://en.wikipedia.org/wiki/Orthogonality
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`cyclooctyne without any cross-reaction, so these are mutually orthogonal reactions, and so, can be performed
`simultaneously and selectively.[15] Bioorthogonal chemistry refers to chemical reactions occurring inside living
`systems without reacting with naturally present cellular components. In supramolecular chemistry the notion
`of orthogonality refers to the possibility of two or more supramolecular, often non-covalent, interactions being
`compatible; reversibly forming without interference from the other.
`
`In analytical chemistry, analyses are "orthogonal" if they make a measurement or identification in completely
`different ways, thus increasing the reliability of the measurement. This is often required as a part of a new drug
`application.
`
`System reliability
`In the field of system reliability orthogonal redundancy is that form of redundancy where the form of backup
`device or method is completely different from the prone to error device or method. The failure mode of an
`orthogonally redundant back-up device or method does not intersect with and is completely different from the
`failure mode of the device or method in need of redundancy to safeguard the total system against catastrophic
`failure.
`
`Neuroscience
`In neuroscience, a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality)
`is called an orthogonal map.
`
`Gaming
`In board games such as chess which feature a grid of squares, 'orthogonal' is used to mean "in the same
`row/'rank' or column/'file'". This is the counterpart to squares which are "diagonally adjacent".[16] In the
`ancient Chinese board game Go a player can capture the stones of an opponent by occupying all orthogonally-
`adjacent points.
`
`Other examples
`Stereo vinyl records encode both the left and right stereo channels in a single groove. The V-shaped groove in
`the vinyl has walls that are 90 degrees to each other, with variations in each wall separately encoding one of
`the two analogue channels that make up the stereo signal. The cartridge senses the motion of the stylus
`following the groove in two orthogonal directions: 45 degrees from vertical to either side.[17] A pure horizontal
`motion corresponds to a mono signal, equivalent to a stereo signal in which both channels carry identical (in-
`phase) signals.
`
`See also
`◾ Imaginary number
`◾ Isogonal
`◾ Isogonal trajectory
`
`https://en.wikipedia.org/wiki/Orthogonality
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`◾ Orthogonal complement
`◾ Orthogonal group
`◾ Orthogonal matrix
`◾ Orthogonal polynomials
`◾ Orthogonalization
`◾ Gram–Schmidt process
`◾ Orthonormal basis
`◾ Orthonormality
`◾ Orthogonal transform
`◾ Pan-orthogonality occurs in coquaternions
`◾ Surface normal
`◾ Orthogonal ligand-protein pair
`
`Notes
`1. Liddell and Scott, A Greek–English Lexicon s.v. ὀρθός (http://www.perseus.tufts.edu/hopper/morph?l=o%
`29rqos&la=greek#lexicon)
`2. Liddell and Scott, A Greek–English Lexicon s.v. γωνία (http://www.perseus.tufts.edu/hopper/morph?
`l=gwni%2Fa&la=greek#lexicon)
`3. Liddell and Scott, A Greek–English Lexicon s.v. ὀρθογώνιον (http://www.perseus.tufts.edu/hopper/morph?
`l=o%29rqog%2Fwnion&la=greek#lexicon)
`4. Oxford English Dictionary, Third Edition, September 2004, s.v. orthogonal
`5. J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 58. ISBN 0-7167-
`0344-0.
`6. "Wolfram MathWorld" (http://mathworld.wolfram.com/Orthogonal.html).
`7. Bourbaki, "ch. II §2.4", Algebra I, p. 234
`8. Trefethen, Lloyd N. & Bau, David (1997). Numerical linear algebra (https://books.google.com/books?id=bj-
`Lu6zjWbEC&pg=PA13). SIAM. p. 13. ISBN 978-0-89871-361-9.
`9. R. Penrose (2007). The Road to Reality. Vintage books. pp. 417–419. ISBN 0-679-77631-1.
`10. Michael L. Scott, Programming Language Pragmatics, p. 228
`11. 1968, Adriaan van Wijngaarden et al., Revised Report on the Algorithmic Language ALGOL 68, section
`0.1.2, Orthogonal design
`12. Null, Linda & Lobur, Julia (2006). The essentials of computer organization and architecture
`(https://books.google.com/books?id=QGPHAl9GE-IC&pg=PA257) (2nd ed.). Jones & Bartlett Learning.
`p. 257. ISBN 978-0-7637-3769-6.
`13. Athanasios Papoulis; S. Unnikrishna Pillai (2002). Probability, Random Variables and Stochastic
`Processes. McGraw-Hill. p. 211. ISBN 0-07-366011-6.
`14. Hedayat, A.; et al. (1999). Orthogonal arrays: theory and applications (https://books.google.com/books?
`id=HrUYlIbI2mEC&pg=PA168). Springer. p. 168. ISBN 978-0-387-98766-8.
`
`https://en.wikipedia.org/wiki/Orthogonality
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`15. Karver, Mark R.; Hilderbrand, Scott A. (2012). "Bioorthogonal Reaction Pairs Enable Simultaneous,
`Selective, Multi-Target Imaging" (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3304098). Angewandte
`Chemie International Edition. 51 (4): 920–2. doi:10.1002/anie.201104389
`(https://doi.org/10.1002/anie.201104389). PMC 3304098
`(https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3304098)
`(https://www.ncbi.nlm.nih.gov/pubmed/22162316).
`16. "chessvariants.org chess
`glossary" (http://www.chessvariants.org/misc.dir/coreglossary.html#orthogonal_direction).
`17. For an illustration, see YouTube (https://www.youtube.com/watch?v=umu37m0qUiE).
`
`. PMID 22162316
`
`References
`◾ Chapter 4 – Compactness and Orthogonality (http://www.faqs.org/docs/artu/ch04s02.html) in The Art of
`Unix Programming
`
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