orthogonal, adj.
Pronunciation: Brit. /ɔːˈθɒɡənl/, U.S. /ɔrˈθɑɡən(ə)l/
Forms: 15–16 orthogonall, 16 18– orthogonal.
Etymology: < Middle French, French orthogonal having a right angle (1520) < orthogone (see orthogon n.) + -al -al suffix1. Compare post-classical Latin orthogonalis (c1109, a1336 in British sources). Compare later orthogon n.
1.
a. Designating a right angle; (also) having a right angle; right-angled; rectangular. Now rare.
?a1560 L. Digges Geom Pract.: Pantometria (1571) i. Defs. sig. B j v, Of straight lined angles there are three kindes, the Orthogonall, the Obtuse and the Acute Angle.
?a1560 L. Digges Geom. Pract.: Pantometria (1571) i. Defs. sig. B j v, Eche of those Angles is an Orthogonall or right Angle.
1611 R. Cotgrave Dict. French & Eng. Tongues, Orthogonal, orthogonall, right cornered.
1612 J. Selden in M. Drayton Poly-olbion i. To Rdr. sig. A3, Pythagoras his sacrifice after his Geometricall Theorem in finding the Squares of an Orthogonall Triangles sides.
1871 L. Colange Zell's Pop Encycl. II. 531/2 Orthogonal..Right-angled; rectangular.
1979 W. Styron Sophie's Choice (2000) 327 The old man..seemed thoroughly exhilarated—the clean bright orthogonal Mondrians bringing special delight to his technician's eye.
b. Relating to or involving right angles; at right angles (to something else).
1694 P. A. Motteux Wks. F. Rabelais (1737) v. 235 An Orthogonal Line.
1878 H. P. Gurney Crystallogr. 37 If two symmetral planes intersect at right angles the line in which they cut is called an axis of orthogonal symmetry.
1892 A. E. H. Love Treat. Math. Theory Elasticity I. 200 Systems of surfaces which cut each other everywhere at right angles. Such systems are called orthogonal surfaces.
1953 S. W. Amos & D. C. Birkinshaw Telev. Engin. I. iv. 107 It is important, therefore, that the scanning beam should approach the target strictly at right angles at all points; in other words orthogonal scanning is necessary.
1978 P. W. Atkins Physical Chem. xv. 476 Two orthogonal (perpendicular) σ bonds are formed in this way.
1989 B. Alberts et al. Molecular Biol. Cell (ed. 2) iv. 171 The narrow gel containing the separated proteins is again subjected to electrophoresis but in a direction orthogonal to that used in the first step.
2. Math.
a. Of a linear transformation of a vector space: preserving lengths and angles; leaving unchanged the inner product of any two vectors.
1852 J. J. Sylvester in Cambr. & Dublin Math. Jrnl. 7 57 Then x2 + y2 + z2 remains invariable, which is the well-known characteristic of orthogonal transformation.
1893 L. G. Weld Theory of Determinants ix. 227 The transformation, in analytical geometry, from one set of axes to another, without changing the origin, is orthogonal.
1941 G. Birkhoff & S. MacLane Surv. Mod. Algebra ix. 222 A linear transformation T is orthogonal if it preserves the absolute value of every vector ξ, so that |ξT| = |ξ|.
1972 F. E. Hohn Introd. Linear Algebra viii. 237 An orthogonal transformation of ℰn maps orthogonal vectors onto orthogonal vectors and nonorthogonal vectors onto nonorthogonal vectors.
1998 R. H. Heck in G. A. Marcoulides Mod. Methods Business Res. vii. 195 The goal of orthogonal rotation is to maintain the 90° of separation between the clusters of variables but to get each set of variables clustered on an axis.
b. 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.
1891 Proc. London Math. Soc. 22 449 (title) On certain properties of symmetric, skew symmetric, and orthogonal matrices.
1907 M. Bôcher Introd. Higher Algebra xi. 154 An orthogonal transformation. [Note] The matrix of such a transformation is called an orthogonal matrix.
1964 N. N. Hancock Matrix Anal. Electr. Machinery ii. 18 The value of the determinant of an orthogonal matrix is necessarily ± 1, but the converse is not true.
1990 IMA Jrnl Numerical Anal. 10 551 Θk is an m × m orthogonal matrix corresponding to a suitable Householder transformation (or a product of Givens transformations).
c. Designating the group of all orthogonal matrices of a given order.
1898 Bull. Amer. Math. Soc. 4 196 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 +…+ ξm2... The order of the orthogonal group G on m indices in the GF[2n] is thus [etc.].
1941 G. Birkhoff & S. MacLane Surv. 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.
1972 F. E. Hohn Introd. Linear Algebra viii. 252 Show that the set of all linear operators on ℰn of the form Y = UX, where U is orthogonal, constitute a group (the orthogonal group).
1992 G. Ellis Rings & Fields viii. 138 The orthogonal group..consisting of all n × n complex matrices which are orthogonal
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 elements is zero if and only if the two are distinct. Also: involving such factors or functions.
1913 Proc. London Math. Soc. 12 297 The theory of Fourier series and of other series of orthogonal functions.
1926 E. W. Hobson Theory of Functions of Real Variable (ed. 2) II. x. 754 If {ψn(x)} be a complete sequence of linearly independent functions for the interval (a, b), a normal orthogonal and complete system of functions {ϕn(x)} can be so determined that ϕn(x) is a linear function of ψ1(x), ψ2(x),…ψn(x).
1941 R. V. Churchill Fourier Series iii. 45 The functions einx = cos nx + i sin nx (n = 0, ±1, ±2,…) form a system which is orthogonal on the interval (−π, π).
1967 A. A. Goldstein Constructive Real Anal. 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.
1967 A. A. Goldstein Constructive Real Anal. iii. 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.
1968 C. 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.
1990 T. Petrie & J. Randall Connections, Definite Forms, & Four-manifolds iii. 35 Thus * preserves harmonic forms and so induces an orthogonal splitting H2 = H+ ⊕ H−.
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. Freq. with to.
1933 Jrnl. Agric. Sci. 23 110 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.
1950 M. H. Quenouille Introd. Statistics iv. 59 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.
1973 Jrnl. Genetic Psychol. 122 45 Implicit in the work..is the concept that creativity and intelligence are relatively orthogonal (i.e., unrelated statistically) at high levels of intelligence.
1982 Lang. & Communication 2 261 Orthogonal comparisons between levels of attribute type were made using Wilcoxon's tests.
1991 Forestry 64 364 Each additional new variable accounts for the largest amount of remaining variation such that it is independent of (orthogonal to) the previously derived principal components.
Special uses
orthogonal projection n. a method of projection employing rays which are at right angles to the plane of projection; (also) a projection drawn or made by such means.
1835 Philos. Trans. (Royal Soc.) 125 223 We assume, for reasons too well known to need specification here, that the orthogonal projection of the dipping-needle upon the horizontal plane gives the position of the horizontal needle.
1878 R T. H. Bartley tr. P. Topinard Anthropol. ii. iii. 264 Orthogonal projections are the only ones which give exact measurements applicable to craniometry.
1957 J. Emmons tr. O Benesch Rembrandt 69 The Night Watch..shows Rembrandt laying greater stress on the plane of relief, which he boldly cuts through by means of orthogonal projection, with the result that the plane surface of the canvas is seemingly transformed into three-dimensional space.
orthogonal trajectory n. Geom. a curve intersecting each of a family of curves at right angles.
1816 tr. S. F. Lacroix Elem. Treat. Differential & Integral Calculus 403 The trajectories in which the angle TMt is a right angle, are called orthogonal trajectories.
1882 S. H. Vines tr. F. G. J. von Sachs Text-bk. Bot. 951 The planes of the walls in a growing-point are classified thus: a. Periclinal... b. Anticlinal, those which intersect the surface and the periclinal walls at right angles... If the outline..of the growing-point is a parabola..the anticlinals being the orthogonal trajectories of the periclinals, constitute a system of confocal parabolas.
1904 Trans. Amer. Math. Soc. 5 56 Such a system of geodesics together with the orthogonal trajectories may be regarded as dividing the surface into infinitesimal squares.
1993 Amer. Math. Monthly 100 127 But because z → z2 is conformal, the image H must be an orthogonal trajectory through the new family of ellipsis.
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