Orthonormalis

Orthonormalis is a term used in linear algebra to denote a system of vectors that are orthogonal to one another and of unit length within an inner product space. Formally, a set {v1, ..., vk} in an inner product space V (real or complex) is orthonormal if the inner product satisfies <vi, vj> = 0 for i ≠ j and <vi, vi> = 1 for all i. If the vectors span a subspace, the set is an orthonormal set; if it spans the entire space and is linearly independent, it is an orthonormal basis.

Orthonormal systems simplify many computations: the coordinates of a vector relative to an orthonormal basis are the inner products with the basis vectors. The projection of a vector onto the subspace spanned by an orthonormal set is the sum of the inner products with the basis vectors times those vectors. In real spaces, the inner product is the dot product; in complex spaces, one uses the conjugate transpose, so <vi, vj> = δij.

Common ways to obtain an orthonormal set include the Gram–Schmidt process, which converts a linearly independent set into an orthonormal one, and QR decomposition in which the Q matrix has orthonormal columns. The notion is central to the spectral theorem, Fourier analysis, and many numerical methods.

Examples: In R^n with the usual dot product, the standard basis e1, ..., en is orthonormal. The set of normalized eigenvectors of a symmetric matrix forms an orthonormal basis in its eigenspace.

See also: orthogonal, inner product, Gram–Schmidt, QR decomposition, orthonormal basis, Hilbert space, Fourier basis.