Encyclopedia > Orthonormal

  Article Content

Orthonormal

Two vectors <math>\mathbf{v}</math> and <math>\mathbf{w}</math> are said to be orthonormal if they are both orthogonal and normalized[?]. A set of vectors which are pairwise orthonormal is called an orthonormal set.

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<\phi(x),\psi(x)>\ = \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>

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

See also: normalized vector, orthogonal, Lp space.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Brazil

... Inácio Lula da Silva Area - Total  - % waterRanked 5th 8,511,965 km² 0.65% Population  - Total (2000)  - DensityRanked ...

 
 
 
This page was created in 35.3 ms