orthogonal matrices
A real square matrix is orthogonal if , i.e., if . The rows and columns of an orthogonal matrix form an orthonormal basis.
Orthogonal matrices play a very important role in linear algebra. Inner products are preserved under an orthogonal transform: , and also the Euclidean norm . An example of where this is useful is solving the least squares problem by solving the equivalent problem .
Orthogonal matrices can be thought of as the real case of unitary matrices. A unitary matrix has the property , where (the conjugate transpose). Since for real , orthogonal matrices are unitary.
An orthogonal matrix has .
Important orthogonal matrices are Givens rotations and Householder transformations. They help us maintain numerical stability because they do not amplify rounding errors.
Orthogonal matrices are rotations or reflections if they have the form:
respectively.
This entry is based on content from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)
References
- 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.
Title | orthogonal matrices |
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Canonical name | OrthogonalMatrices |
Date of creation | 2013-03-22 12:05:19 |
Last modified on | 2013-03-22 12:05:19 |
Owner | akrowne (2) |
Last modified by | akrowne (2) |
Numerical id | 11 |
Author | akrowne (2) |
Entry type | Definition |
Classification | msc 15-00 |
Related topic | OrthogonalPolynomials |
Related topic | RotationMatrix |