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orthogonal, a. Geom.|ɔːˈθɒgənəl|
[a. F. orthogonal, f. orthogone: see prec. and -al1; and cf. hexagonal, etc.]
1. Having or of the nature of a right angle, right-angled (obs.); pertaining to or involving right angles; at right angles to something else, or to each other; rectangular.
orthogonal projection, projection in which the rays are at right angles to the plane of projection. orthogonal trajectory, a curve intersecting each of a family of curves at right angles.
1571Digges Pantom. i. Elem. B j b, Of straight lined angles there are three kindes, the Orthogonall, the Obtuse and the Acute Angle. Ibid., Eche of those Angles is an Orthogonall or right Angle. 1612Selden in Illustr. Drayton's Poly-olb. A iij, Pythagoras's sacrifice after his Geometricall Theorem in finding the squares of an Orthogonall triangles sides. 1694Motteux Rabelais v. (1737) 235 An Orthogonal Line. 1816tr. Lacroix's Diff. & Int. Calculus 403 The trajectories in which the angle TMt is a right angle, are called orthogonal trajectories. 1878Gurney Crystallogr. 37 If two symmetral planes intersect at right angles the line in which they cut is called an axis of orthogonal symmetry. 1878Bartley tr. Topinard's Anthrop. ii. iii, Orthogonal projections are the only ones which give exact measurements applicable to craniometry.
2. Math.
a. Of a linear transformation: preserving lengths and angles; leaving unchanged quantities of the form x12 + x22 +{ddd}+ xn2 and the inner product of any two vectors.
1859G. Salmon Lessons Introd. Mod. Higher Algebra xv. 125 What we may call the orthogonal transformation is to transform simultaneously a given quadratic function, and x2 + y2 + z2 + w2 + &., so that the latter remaining of the same form, the former may become Ax2 + By2 + Cz2 + Dw2 + &. 1893L. G. Weld Short Course Theory Determinants ix. 227 The transformation, in analytical geometry, from one set of axes to another, without changing the origin, is orthogonal. 1941Birkhoff & MacLane Survey Mod. Algebra ix. 222 A linear transformation T is orthogonal if it preserves the absolute value of every vector ξ, so that {vb}ξT{vb} = {vb}ξ{vb}. 1972F. E. Hohn Introd. Linear Algebra viii. 237 An orthogonal transformation of {scrE}n maps orthogonal vectors onto orthogonal vectors and nonorthogonal vectors onto nonorthogonal vectors.
b. Applied to the group of all orthogonal matrices of a given order.
1898Bull. Amer. Math. Soc. IV. 196 A linear substitution S on the marks of a Galois Field of order pn..will be called orthogonal if it leaves absolutely invariant ξ12 + ξ22 +{ddd}+ ξm2... The order of the orthogonal group G on m indices in the GF[2n] is thus [etc.]. 1941Birkhoff & MacLane Survey Mod. Algebra ix. 225 This subgroup of the full linear group..is called the orthogonal group On; it is isomorphic to the group of all orthogonal transformations of the given Euclidean space. 1972F. E. Hohn Introd. Linear Algebra viii. 252 Show that the set of all linear operators on {scrE}n of the form Y = UX, where U is orthogonal, constitute a group (the orthogonal group).
c. Of a square matrix: representing an orthogonal transformation; such that the rows (and likewise the columns) are orthonormal when considered as vectors; equal to the inverse of its transpose; (these three properties are equivalent).
1907M. Bôcher Introd. Higher Algebra xi. 154 An orthogonal transformation. [Note] The matrix of such a transformation is called an orthogonal matrix. 1964N. N. Hancock Matrix Anal. Electr. Machinery ii. 18 The value of the determinant of an orthogonal matrix is necessarily {pm} 1, but the converse is not true.
d. Of two vectors or functions: perpendicular; having an inner product equal to zero. Of a set of vectors or functions: such that the inner product of any two is zero if and only if the two are distinct.
1913Proc. London Math. Soc. XII. 297 The theory of Fourier series and of other series of orthogonal functions. 1926E. W. Hobson Theory of Functions of Real Variable (ed. 2) II. x. 754 If {ob}ψn(x){cb} be a complete sequence of linearly independent functions for the interval (a, b), a normal orthogonal and complete system of functions {ob}ϕn(x){cb} can be so determined that ϕn(x) is a linear function of ψ1(x), ψ2(x),{ddd}ψn(x). 1941R. V. Churchill Fourier Series iii. 45 The functions einx = cos nx + i sin nx (n = 0, {pm}1, {pm}2,{ddd}) form a system which is orthogonal on the interval (-π, π). 1967A. A. Goldstein Constructive Real Analysis iii. 112 We define an inner product space I[a, b] by introducing an inner product..defined by [f, g] = ∫ba f(t) g(t)dt. Two functions f and g in I[a, b] are said to be orthogonal if [f, g] = 0. Ibid. 115 Two points x and y of [a Hilbert space] H are orthogonal if [x, y] = 0. Similarly, two subspaces M and N of H are said to be orthogonal if [M, N] = 0. 1968C. G. Kuper Introd. Theory Superconductivity i. 3 Bardeen, Cooper and Schrieffer (1957) constructed a variational wave function for a ground state with complete electron pairing, and orthogonal functions for low-lying excited states having only a few such pairs broken.
3. Statistics. Of a set of variates: statistically independent. Of an experimental design: such that the variates under investigation can be treated as statistically independent.
1933Jrnl. Agric. Sci. XXIII. 110 In an ordinary replicated field experiment of the randomised block or Latin square type the differences of the means of plots receiving the same treatments are taken without hesitation to be true measures of treatment differences, but this is only so because the experiment has been specially arranged so as to be orthogonal. 1950M. H. Quenouille Introd. Statistics iv. 59 If we are comparing a series of..measurements on people to determine the effect of age, these comparisons may be complicated by the effect of sex... The only manner in which we can assume that sex does not enter into the comparison is to choose the same proportion of each sex in each age group. The effect of sex is then said to be ‘orthogonal’ to the effect of age. 1967Word XXIII. 219 Another model which provides a relevant comparison to phonological distinctive features is the mathematical method of factor analysis... The various mathematical methods employed lead to the positing of a number of independent ‘orthogonal’ factors and each test or other set of responses is described in terms of positive or negative loadings on each factor. 1973Jrnl. Genetic Psychology CXXII. 45 Implicit in the work..is the concept that creativity and intelligence are relatively orthogonal (i.e., unrelated statistically) at high levels of intelligence. |