Orthonormality

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In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (their inner product is 0) and both of unit length (the norm of each is 1). A set of vectors which is pairwise orthonormal (any two vectors in it are orthonormal) is called an orthonormal set. A basis which forms an orthonormal set is called an orthonormal basis.

For example, the standard basis for Euclidean 3-space {i,j,k} is orthonormal, because i·j = 0, j·k = 0, k·i = 0 and each of them is a unit vector.

A set of vectors can be transformed into an orthonormal set by applying the Gram–Schmidt process, and then normalizing each vector.


When referring to real-valued functions, usually the L² inner product is assumed unless otherwise stated, so 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 Kronecker delta. A set of vectors (functions, matrices, sequences etc)

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

forms an orthonormal set if and only if

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

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