Encyclopedia > Orthonormal

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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.

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