Home > Orthogonality

1 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.

2 Orthogonal functions

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

Here we introduce a nonnegative weight function , and we write

.

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

• orthonormal

where

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.

3 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:

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,

• The functions 1, sin(nx), cos(nx) : n = 1, 2, 3, ... are orthogonal with respect to Lebesgue measureMeasure theory The Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets which can be assigned a volume are called Lebesgue measura on the interval from 0 to 2π. This fact is basic in the theory of Fourier seriesIn mathematics, a Fourier series named in honor of Joseph Fourier ( 1768- 1830), is a representation of a periodic function (often taken to have period 2π in a sense, the simplest case) as a sum of periodic functions of the form : which are harmonics o.

• Various eponymously named polynomial sequences are sequences of orthogonal polynomials. In particular:
  • The Hermite polynomialsIn mathematics, the Hermite polynomials named in honor of Charles Hermite (pronounced "air MEET"), are a polynomial sequence defined either by : (the probabilists' Hermite polynomials , or sometimes by : (the physicists' Hermite polynomials . These two de are orthogonal with respect to the normal distributionProbability density function of Gaussian distribution (bell curve). The normal distribution is an extremely important probability distribution in many fields. It is also called the Gaussian distribution especially in physics and engineering. It is actuall with expected value 0.
  • The Legendre polynomialsNote: The term Legendre polynomials is sometimes used (wrongly) to indicate the associated Legendre polynomials. In mathematics, Legendre functions are solutions to Legendre's differential equation : They are named after Adrien-Marie Legendre. This ordina are orthogonal with respect to the uniform distributionIn mathematics, the uniform distributions are simple probability distributions. The distribution can be either discrete or continuous. In the discrete case, they can be characterized by saying that all possible values are equally probable. In the continuo on the interval from −1 to 1.
  • The Laguerre polynomialsIn mathematics, the Laguerre polynomials named after Edmond Laguerre (1834 1886), are a polynomial sequence defined by : These polynomials are orthogonal to each other with respect to the inner product given by : Generalization The orthogonality property 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
  • The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution.