Uniform random orthogonal matrix

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This section considers the generation of uniform random matrices $Q in RR^{n xx n}$ such that $Q^T Q = I$ and $det(Q) = +- 1$ . In other words, we would like to generate a random matrix $Q in O(n)$ , such that the distribution of $Q$ is given by the Haar measure $P$ :

$forall Gamma, Q in O(n): P(Q in A) = P(Q in Gamma A),$

for all measurable $A subset O(n)$ .

QR decomposition

The QR-decomposition is not unique. If

$M = QR$

is a QR-decomposition, then also

$M = (Q D) (D^-1 R)$

is a QR-decomposition when $D$ is a diagonal orthogonal matrix (diagonal consists of $+-1$ ). To make the QR-decomposition unique, the signs of the diagonal of $R$ must be fixed. For example, to make the diagonal of $R$ positive, $D$ is chosen as

$D = diag(sign(diag(R))).$

If $M in RR^{n xx n}$ is generated from i.i.d. standard-normal real random variables, then $QD$ is distributed with the Haar measure in $O(n)$ .

References

How to Generate Random Matrices from the Classical Compact Groups, Francesco Mezzadri, Notices of the AMS, Volume 54, Number 5, 2007.

Files

Uniform random orthogonal matrix

random_orthogonal_matrix.h