Editing Orthogonal matrix (section)

===Randomization===
Some numerical applications, such as [[Monte Carlo method]]s and exploration of high-dimensional data spaces, require generation of [[uniform distribution (continuous)|uniformly distributed]] random orthogonal matrices. In this context, "uniform" is defined in terms of [[Haar measure]], which essentially requires that the distribution not change if multiplied by any freely chosen orthogonal matrix. Orthogonalizing matrices with [[statistical independence|independent]] uniformly distributed random entries does not result in uniformly distributed orthogonal matrices{{Citation needed|date=June 2009}}, but the [[QR decomposition|{{mvar|QR}} decomposition]] of independent [[normal distribution|normally distributed]] random entries does, as long as the diagonal of {{mvar|R}} contains only positive entries {{harv|Mezzadri|2006}}. {{harvtxt|Stewart|1980}} replaced this with a more efficient idea that {{harvtxt|Diaconis|Shahshahani|1987}} later generalized as the "subgroup algorithm" (in which form it works just as well for permutations and rotations). To generate an {{math|(''n'' + 1) × (''n'' + 1)}} orthogonal matrix, take an {{math|''n'' × ''n''}} one and a uniformly distributed unit vector of dimension {{nowrap|''n'' + 1}}. Construct a Householder reflection from the vector, then apply it to the smaller matrix (embedded in the larger size with a 1 at the bottom right corner).