orthogonal matrices

A real square n×n matrix Q is orthogonalMathworldPlanetmathPlanetmath if QTQ=I, i.e., if Q-1=QT. The rows and columns of an orthogonal matrixMathworldPlanetmath form an orthonormal basisMathworldPlanetmath.

Orthogonal matrices play a very important role in linear algebra. Inner productsMathworldPlanetmath are preserved under an orthogonal transform: (Qx)TQy=xTQTQy=xTy, and also the Euclidean normMathworldPlanetmath ||Qx||2=||x||2. An example of where this is useful is solving the least squares problem Axb by solving the equivalent problem QTAxQTb.

Orthogonal matrices can be thought of as the real case of unitary matricesMathworldPlanetmath. A unitary matrix Un×n has the property U*U=I, where U*=UT¯ (the conjugate transposeMathworldPlanetmath). Since QT¯=QT for real Q, orthogonal matrices are unitary.

An orthogonal matrix Q has det(Q)=±1.

Important orthogonal matrices are Givens rotations and Householder transformations. They help us maintain numerical stability because they do not amplify rounding errors.

Orthogonal 2×2 matrices are rotationsMathworldPlanetmath or reflections if they have the form:



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)


  • 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.
Title orthogonal matrices
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