Definition

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

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

If $x,y \in S$ and $x\ne y$ , then is orthogonal to .
If $x \in S$ , then the norm of is .

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.

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orthonormal

Attachments:: every orthonormal set is linearly independent (Theorem) by mathcam

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Cross-references: simple, Gram-Schmidt orthonormalization, vector space, orthonormal basis, orthogonal matrix, real, empty set, vectors, norm, Kronecker delta, inner product, inner product space, subset
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This is version 10 of orthonormal set, born on 2002-01-04, modified 2007-01-08.
Object id is 1283, canonical name is Orthonormal.
Accessed 14889 times total.

Classification:

AMS MSC:

65F25 (Numerical analysis :: Numerical linear algebra :: Orthogonalization)

Pending Errata and Addenda

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