Orthogonal matrices

A real square (n,n) matrix is orthogonal if File:Hepa img803.gif , i.e. if

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. Orthogonal matrices play a very important role in linear algebra. Inner products are preserved under an orthogonal transform: File:Hepa img804.gif , and of course the Euclidean norm File:Hepa img805.gif , so that we can, e.g. solve the least squares problem

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by solving the equivalent problem File:Hepa img806.gif .

Important examples are Givens rotations Householder transformations. They will help us to maintain numerical stability because they do not amplify rounding errors.

Orthogonal (2,2) matrices are rotations or reflections if they have the form:

respectively.