Editing Orthogonal matrix (section)

===Benefits===
[[Numerical analysis]] takes advantage of many of the properties of orthogonal matrices for numerical linear algebra, and they arise naturally. For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonal change of bases; both take the form of orthogonal matrices. Having determinant ±1 and all eigenvalues of magnitude 1 is of great benefit for [[numeric stability]]. One implication is that the [[condition number]] is 1 (which is the minimum), so errors are not magnified when multiplying with an orthogonal matrix. Many algorithms use orthogonal matrices like Householder reflections and [[Givens rotation]]s for this reason. It is also helpful that, not only is an orthogonal matrix invertible, but its inverse is available essentially free, by exchanging indices.

Permutations are essential to the success of many algorithms, including the workhorse [[Gaussian elimination]] with [[Pivot element#Partial and complete pivoting|partial pivoting]] (where permutations do the pivoting). However, they rarely appear explicitly as matrices; their special form allows more efficient representation, such as a list of {{mvar|n}} indices.

Likewise, algorithms using Householder and Givens matrices typically use specialized methods of multiplication and storage. For example, a Givens rotation affects only two rows of a matrix it multiplies, changing a full [[matrix multiplication|multiplication]] of order {{math|''n''<sup>3</sup>}} to a much more efficient order {{mvar|n}}. When uses of these reflections and rotations introduce zeros in a matrix, the space vacated is enough to store sufficient data to reproduce the transform, and to do so robustly. (Following {{harvtxt|Stewart|1976}}, we do ''not'' store a rotation angle, which is both expensive and badly behaved.)