Orthonormal Basis (original) (raw)

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A subset {v_1,...,v_k} of a vector space V, with the inner product <,>, is called orthonormal if <v_i,v_j>=0 when i!=j. That is, the vectors are mutually perpendicular. Moreover, they are all required to have length one: <v_i,v_i>=1.

An orthonormal set must be linearly independent, and so it is a vector basis for the space it spans. Such a basis is called an orthonormal basis.

The simplest example of an orthonormal basis is the standard basis e_i for Euclidean space R^n. The vector e_i is the vector with all 0s except for a 1 in the ith coordinate. For example, e_1=(1,0,...,0). A rotation (or flip) through the origin will send an orthonormal set to another orthonormal set. In fact, given any orthonormal basis, there is a rotation, or rotation combined with a flip, which will send the orthonormal basis to the standard basis. These are precisely the transformations which preserve the inner product, and are called orthogonal transformations.

Usually when one needs a basis to do calculations, it is convenient to use an orthonormal basis. For example, the formula for a vector space projection is much simpler with an orthonormal basis. The savings in effort make it worthwhile to find an orthonormal basis before doing such a calculation.Gram-Schmidt orthonormalization is a popular way to find an orthonormal basis.

Another instance when orthonormal bases arise is as a set of eigenvectors for a symmetric matrix. For a general matrix, the set of eigenvectors may not be orthonormal, or even be a basis.

See also

Complete Orthogonal System, Dot Product, Inner Product,Kronecker Delta, Lie Group, Lorentzian Inner Product, Matrix, Orthogonal Basis,Orthogonal Matrix, Orthogonal Group, Orthogonal Transformation,Orthonormal Functions, Orthonormal Set, Orthonormal Vectors, Symmetric Quadratic Form, Vector Basis, Vector Space Projection

This entry contributed by Todd Rowland

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Rowland, Todd. "Orthonormal Basis." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/OrthonormalBasis.html

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