AskDefine | Define orthogonal

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

Etymology

Mediaeval orthogonalis, from Latin orthogonius "right-angled".

Pronunciation

  • /ɔ:'θɒgənəl/

Adjective

orthogonal (non-comparable)
  1. pertaining to right angles; perpendicular (to)
    A chord and the radius that bisects it are orthogonal.
  2. mathematical use
    1. Of two functions, linearly independent; having a zero inner product.
      The normal vector and tangent vector at a given point are orthogonal.
    2. Of a square matrix that is the inverse of its transpose
    3. Of a linear transformation that preserves angles
  3. statistically independent, with reference to variates
  4. 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
mathematical term
statistically independent
able to be treated separately

French

Etymology

Mediaeval Latin orthogonalis, from late Latin orthogonius "right-angled".

Pronunciation

/ɔʀtɔgɔnal/

Adjective

  1. orthogonal

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.
This means that T preserves the angle between x and y, and that the lengths of Tx and x are equal.
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_
where
\delta_=\left\

Synonyms, Antonyms and Related Words

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