Dictionary Definition
orthogonal adj
1 not pertinent to the matter under
consideration; "an issue extraneous to the debate"; "the price was
immaterial"; "mentioned several impertinent facts before finally
coming to the point" [syn: extraneous, immaterial, impertinent]
2 statistically unrelated
3 having a set of mutually perpendicular axes;
meeting at right angles; "wind and sea may displace the ship's
center of gravity along three orthogonal axes"; "a rectangular
Cartesian coordinate system" [syn: rectangular]
User Contributed Dictionary
English
Pronunciation
- /ɔ:'θɒgənəl/
Adjective
orthogonal (non-comparable)- pertaining to right angles;
perpendicular (to)
- A chord and the radius that bisects it are orthogonal.
- mathematical use
- Of two functions,
linearly independent; having a zero inner
product.
- The normal vector and tangent vector at a given point are orthogonal.
- Of a square matrix that is the inverse of its transpose
- Of a linear transformation that preserves angles
- Of two functions,
linearly independent; having a zero inner
product.
- statistically independent, with reference to variates
- Able to be treated separately.
- The content of the message should be orthogonal to the means of its delivery.
Derived terms
Translations
of right angles
- Dutch: orthogonaal, loodrecht
- Finnish: ortogonaalinen, suorakulmainen
- French: orthogonal
- German: rechtwinklig
- Hungarian: ortogonális
- Swedish: ortogonal, vinkelrät, rätvinklig
mathematical term
- Dutch: orthogonaal
- French: orthogonal
- German: orthogonal
- Hungarian: ortogonális
- Swedish: ortogonal
statistically independent
- French: orthogonal
- Swedish: ortogonal
able to be treated separately
- French: indépendant
French
Pronunciation
/ɔʀtɔgɔnal/Adjective
Extensive Definition
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
(orthos), meaning "straight", and (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 x and y in an inner product space V are orthogonal if their inner product \langle x, y \rangle is zero. This situation is denoted x \perp y.Two vector
subspaces A and B of vector space
V are called orthogonal subspaces if each vector in A is orthogonal
to each vector in B. The largest subspace that is orthogonal to a
given subspace is its orthogonal
complement.
A linear
transformation T : V \rightarrow V is called an orthogonal
linear transformation if it preserves the inner
product. That is, for all pairs of vectors x and y in the inner
product space V,
- \langle Tx, Ty \rangle = \langle x, y \rangle.
A term
rewriting system is said to be
orthogonal if it is 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, i.e. 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 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.
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:- \langle f, g\rangle_w = \int_a^b f(x)g(x)w(x)\,dx.
Here we introduce a nonnegative weight
function w(x) in the definition of this inner product.
We say that those functions are orthogonal if
that inner product is zero:
- \int_a^b f(x)g(x)w(x)\,dx = 0.
We write the norms
with respect to this inner product and the weight function as
- \|f\|_w = \sqrt
The members of a sequence are:
- orthogonal if
- \langle f_i, f_j \rangle=\int_^\infty f_i(x) f_j(x) w(x)\,dx=\|f_i\|^2\delta_=\|f_j\|^2\delta_
- orthonormal
- \langle f_i, f_j \rangle=\int_^\infty f_i(x) f_j(x) w(x)\,dx=\delta_
- \delta_=\left\
Synonyms, Antonyms and Related Words
cube-shaped, cubed, cubic, cubiform, cuboid, diced, foursquare, normal, oblong, orthodiagonal, orthometric, perpendicular, plumb, plunging, precipitous, quadrangular, quadrate, quadriform, quadrilateral, rectangular, rhombic, rhomboid, right-angle,
right-angled, right-angular, sheer, square, steep, straight-up,
straight-up-and-down, tetragonal, tetrahedral, trapezohedral, trapezoid,
up-and-down