orthonormal set
Definition
An orthonormal set is a subset of an inner product space, such that for all . Here is the inner product, and 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:
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1.
If and , then is orthogonal (http://planetmath.org/OrthogonalVector) to .
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2.
If , 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.
Title | orthonormal set |
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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 |