orthonormal set


An orthonormal setMathworldPlanetmath is a subset S of an inner product spaceMathworldPlanetmath, such that x,y=δxy for all x,yS. Here , is the inner productMathworldPlanetmath, and δ is the Kronecker deltaDlmfPlanetmath.

More verbosely, we may say that an orthonormal set is a subset S of an inner product space such that the following two conditions hold:

  1. 1.

    If x,yS and xy, then x is orthogonalMathworldPlanetmath (http://planetmath.org/OrthogonalVector) to y.

  2. 2.

    If xS, then the norm of x is 1.

Stated this way, the origin of the term is clear: an orthonormal set of vectors is both orthogonal and normalized.


Note that the empty set is orthonormal, as is a set consisting of a single vector of unit norm in an inner product space.

The columns (or rows) of a real orthogonal matrixMathworldPlanetmath form an orthonormal set. In fact, this is an example of an orthonormal basisMathworldPlanetmath.


A standard application is finding an orthonormal basis for a vector spaceMathworldPlanetmath, such as by Gram-Schmidt orthonormalizationMathworldPlanetmath. Orthonormal bases are computationally simple to work with.

Title orthonormal set
Canonical name OrthonormalSet
Date of creation 2013-03-22 12:07:24
Last modified on 2013-03-22 12:07:24
Owner yark (2760)
Last modified by yark (2760)
Numerical id 14
Author yark (2760)
Entry type Definition
Classification msc 65F25
Related topic OrthogonalPolynomials
Related topic OrthonormalBasis
Defines orthonormal