Formally, two vectors and in an Inner Product Space are orthogonal if their inner product is zero. This situation is denoted .
Two subspaces and of are called if each vector in is orthogonal to each vector in . Note however that this does not correspond with the geometric concept of perpendicular planes. The largest subspace that is orthogonal to a given subspace is its Orthogonal Complement .
A Linear Transformation is called an if it preserves the inner product. That is, for all pairs of vectors and in the inner product space ,
:
This means that preserves the angle between and ,
and that the lengths of and 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 is sometimes also used in place of orthogonal. However, ''normal'' can also refer to vectors of unit length. In particular, Orthonormal refers to a collection of vectors that are both orthogonal and of unit length. So the orthogonal usage of the term ''normal'' is often avoided.
For example, in a 2- or 3- Dimension al Euclidean Space , two vectors are orthogonal if their Dot Product is zero, i.e., they make an angle of 90° or π/2 Radian s. Hence orthogonality of vectors is a generalization of the concept of Perpendicular . Also a line through the origin is orthogonal to a plane through the origin if they are perpendicular. Note however that there is no correspondence with regard to perpendicular planes.
In 4D 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''. They are said to be ''orthonormal'' if they are all Unit Vector s. Non-zero pairwise orthogonal vectors are always Linearly Independent .
We commonly use the following inner product to say that two Function s ''f'' and ''g'' are :
:
Here we introduce a nonnegative Weight Function , and we write
:.
We write the Norm s with respect to this inner product and the weight function as
|
| |
\int_{-\infty}^\infty f_i(x) f_j(x) w(x)\,dx=f_i^2\delta_{i,j}=f_j^2\delta_{i,j}</math>
|
: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.
- Take two quadratic functions 2''t'' + 3 and 5''t''2 + ''t'' − 17/9. These functions are orthogonal with respect to a unit weight function on the interval from −1 to 1. The product of these two functions is 10''t''3 + 17''t''2 − 7/9 ''t'' − 17/3, and now,
::
::
::
- The functions 1, sin(''nx''), cos(''nx'') : ''n'' = 1, 2, 3, ... are orthogonal with respect to Lebesgue Measure on the interval from 0 to 2π. This fact is basic in the theory of Fourier Series .
Other meanings of the word ''orthogonal'' evolved from its earlier use in mathematics.
In art the
Perspective imagined lines pointing to the
Vanishing Point are referred to as 'orthogonal lines'.
In
Computer Science , an
Instruction Set is said to be if any instruction 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.
Orthogonality is a system design property which enables the making of complex designs feasible and compact. The aim of an orthogonal design is to guarantee that operations within one of its components neither create nor propagate side-effects to other components. For example a car has orthogonal components and controls, e.g. accelerating the vehicle does not influence anything else but the components involved in the acceleration. On the other hand, a car with non-orthogonal design might have, for example, the acceleration influencing the radio tuning or the display of time. Consequently, this usage is seen to be derived from the use of ''orthogonal'' in mathematics; one may project a vector onto a subspace by projecting it onto each member of a set of
Basis Vectors separately and adding the projections if and only if the basis vectors are mutually orthogonal.
Orthogonality 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. The emergent behaviour 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 the test and development time, because it's easier to verify designs that neither cause side effects nor depend on them.
In radio communications, multiple access schemes are when a receiver can (theoretically) completely reject an arbitrarily strong unwanted signal. An example of an orthogonal scheme is Code Division Multiple Access,
CDMA . Examples of non-orthogonal schemes are
TDMA and
FDMA .
In the social sciences, variables that affect a particular result are said to be orthogonal if they are independent. That is to say that by varying each separately, one can predict the combined effect of varying them jointly. If
Synergistic effects are present, the factors are not orthogonal. This meaning derives from the mathematical one, because orthogonal vectors are linearly independent.
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.
In combinatorics, two ''n''×''n''
Latin Squares are said to be orthogonal if their superimposition yields all possible ''n''
2 combinations of entries. One can also have a
more general definition of combinatorial orthogonality.
