Orthogonality

In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. It means at right angles. It comes from the Greek "ortho", meaning "right" and "gonia", meaning "angle". Two streets that cross each other at a right angle are orthogonal to each other. Two vectors in an inner product space are orthogonal if their inner product is zero. If the vectors are <math>x<math> and <math>y<math> this is written <math>x \perp y<math>. The word normal is sometimes also used for this concept by mathematicians, although that word is rather overburdened.

Contents

1 In Euclidean vector spaces

2 Orthogonal functions

3 Examples

4 Derived meanings

5 Related topics

6 References and external links

In Euclidean vector spaces

For example, in a 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 is a generalization of the concept of perpendicular.

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 vectors. Non-zero pairwise orthogonal vectors are always linearly independent.

Orthogonal functions

We commonly use the following inner product to say that two functions f and g are orthogonal:

<math> \langle f, g \rangle = \int_a^b f(x)g(x)w(x)\,dx = 0.<math>

Here we introduce a nonnegative weight function <math>w(x)<math>, and we write

<math> \langle f, gw \rangle = \langle f, g\rangle_w<math>.

We write the norms with respect to this inner product and the weight function as

The members of a sequence { f_i : i = 1, 2, 3, ... } are:

orthogonal if

<math>\langle f_i, f_j \rangle=\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>

orthonormal

<math>\langle f_i, f_j \rangle=\int_{-\infty}^\infty f_i(x) f_j(x) w(x)\,dx=\delta_{i,j}<math>

where

<math>\delta_{i,j}=\left\{\begin{matrix}1 & \mathrm{if}\ i=j \\ 0 & \mathrm{if}\ i\neq j\end{matrix}\right\}<math>

is Kronecker's delta. In other words, any two of them are orthogonal and the norm of each is 1. See in particular orthogonal polynomials.

Examples

The vectors (1, 3, 2), (3, −1, 0), (1/3, 1, −5/3) are orthogonal to each other, since (1)(3) + (3)(−1) + (2)(0) = 0, (3)(1/3) + (−1)(1) + (0)(−5/3) = 0, (1)(1/3) + (3)(1) − (2)(5/3) = 0. Observe also that the dot product of the vectors with themselves are the norms of those vectors, so to check for orthogonality, we need only check the dot product with every other vector.

The vectors (1, 0, 1, 0, ...)^T and (0, 1, 0, 1, ...)^T are orthogonal to each other. Clearly the dot product of these vectors is 0. We can then make the obvious generalization to consider the vectors in Z₂ⁿ:

<math>\mathbf{v}_k = \sum_{\begin{matrix}i=0\\ai+k < n\end{matrix}}^{n/a} \mathbf{e}_i<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 2t + 3 and 5t² + 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 10t³ + 17t² − 7/9 t − 17/3, and now,

<math>\int_{-1}^{1} \left(10t^3+17t^2-{7\over 9}t-{17\over 3}\right)\,dt = \left[{5\over 2}t^4+{17\over 3}t^3-{7\over 18}t^2-{17\over 3}t\right]_{-1}^{1}<math>

<math>=\left({5\over 2}(1)^4+{17\over 3}(1)^3-{7\over 18}(1)^2-{17\over 3}(1)\right)-\left({5\over 2}(-1)^4+{17\over 3}(-1)^3-{7\over 18}(-1)^2-{17\over 3}(-1)\right)<math>

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.

Various eponymously named polynomial sequences are sequences of orthogonal polynomials. In particular:
- The Hermite polynomials are orthogonal with respect to the normal distribution with expected value 0.
- The Legendre polynomials are orthogonal with respect to the uniform distribution on the interval from −1 to 1.
- The Laguerre polynomials are orthogonal with respect to the exponential distribution. Somewhat more general Laguerre polynomial sequences are orthogonal with respect to gamma distributions.
- The Chebyshev polynomials of the first kind are orthogonal with respect to the measure <math>1/\sqrt{1-x^2}.<math>
- The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution.

Derived meanings

Other meanings of the word orthogonal evolved from its earlier use in mathematics.

In art the perspective lines at an imagined right angle to the picture plane, pointing to the vanishing point are referred to as 'orthogonal lines'.

In computer science, an instruction set is said to be orthogonal 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. As with a set of mathematical basis vectors, which must be orthogonal if they are to represent any vector uniquely, only an orthogonal instruction set can uniquely encode 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 achieve that operations within one of its components do not 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 each member of a set of basis vectors separately and adding the projections if and only if the basis vectors are orthogonal to each other.

Orthogonality achieves that modifying the technical effect produced by a component of a system does not create or propagate 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 orthogonal when a receiver can (theoretically) completely reject an arbitrarily strong unwanted signal. The orthogonal schemes are TDMA and FDMA. A non-orthogonal scheme is Code Division Multiple Access, CDMA.