Orthonormality

From Academic Kids

In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. A set of vectors which are pairwise orthonormal is called an orthonormal set. A basis which forms an orthonormal set is called an orthonormal basis.

When referring to functions, usually the L²-norm is assumed unless otherwise stated, so that two functions <math>\phi(x)<math> and <math>\psi(x)<math> are orthonormal over the interval <math>[a,b]<math> if

<math>(1)\quad\langle\phi(x),\psi(x)\rangle = \int_a^b\phi(x)\psi(x)dx = 0,\quad{\rm and}<math>
<math>(2)\quad||\phi(x)||_2 = ||\psi(x)||_2 = \left[\int_a^b|\phi(x)|^2dx\right]^\frac{1}{2} = \left[\int_a^b|\psi(x)|^2dx\right]^\frac{1}{2} = 1.<math>

An equivalent formulation of the two conditions is done by using the Delta function. A set of vectors (functions, matrices, sequences etc)

<math> \left\{ u_1 , u_2 , ... , u_n , ... \right\} <math>

forms an orthonormal set iff

<math> \forall n,m \ : \quad \left\langle u_n | u_m \right\rangle = \delta_{n,m} <math>

where < | > is the proper inner product defined over the vector space.

Unfortunately, the word normal is sometimes used synonymously with orthogonal.

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