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Encyclopedia > Orthonormal set

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 Lē-norm is assumed unless otherwise stated, so that two functions φ(x) and ψ(x) are orthonormal over the interval [a,b] if

(1)\quad\langle\phi(x),\psi(x)\rangle = \int_a^b\phi(x)\psi(x)dx = 0,\quad{\rm and}
(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.

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

\left\{ u_1 , u_2 , ... , u_n , ... \right\}

forms an orthonormal set iff

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

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


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


  Results from FactBites:
 
Orthonormality - Wikipedia, the free encyclopedia (194 words)
In linear algebra, two vectors v and w in an inner product space are said to be orthonormal if they are orthogonal and of unit length (that is, their inner product is zero and the norm of each of them is 1).
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.
PlanetMath: orthonormal (119 words)
The empty set is unqualifiedly considered an orthonormal set as well.
Orthonormal bases are computationally simple to work with.
This is version 7 of orthonormal, born on 2002-01-04, modified 2003-04-19.
  More results at FactBites »

 

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