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orthonormal set (Definition)

Definition

An orthonormal set is a subset $S$ of an inner product space, such that $\ip{x,y}=\delta_{xy}$ for all $x,y\in S$ Here $\ip{\cdot,\cdot}$ 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 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.




"orthonormal set" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: orthogonal polynomials, orthonormal basis

Also defines:  orthonormal

Attachments:
every orthonormal set is linearly independent (Theorem) by mathcam
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Cross-references: simple, Gram-Schmidt orthonormalization, vector space, application, orthonormal basis, orthogonal matrix, real, empty set, vectors, norm, Kronecker delta, inner product, inner product space, subset
There are 40 references to this entry.

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 18263 times total.

Classification:
AMS MSC65F25 (Numerical analysis :: Numerical linear algebra :: Orthogonalization)

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