orthonormal set

Definition

An orthonormal set is a subset $S$ of an inner product space, such that $⟨x,y⟩={\delta }_{xy}$ for all $x,y\in S$. Here $⟨\cdot ,\cdot ⟩$ is the inner product, and $\delta$ is the Kronecker delta.

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. If $x,y\in S$ and $x\ne y$, then $x$ is orthogonal (http://planetmath.org/OrthogonalVector) to $y$.

2. If $x\in S$, 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.

Notes

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 matrix form an orthonormal set. In fact, this is an example of an orthonormal basis.

Applications

A standard application is finding an orthonormal basis for a vector space, such as by Gram-Schmidt orthonormalization. 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