scipy.stats.ortho_group = <scipy.stats._multivariate.ortho_group_gen object>[source]#

An Orthogonal matrix (O(N)) random variable.

Return a random orthogonal matrix, drawn from the O(N) Haar distribution (the only uniform distribution on O(N)).

The dim keyword specifies the dimension N.


Dimension of matrices

seed{None, int, np.random.RandomState, np.random.Generator}, optional

Used for drawing random variates. If seed is None, the RandomState singleton is used. If seed is an int, a new RandomState instance is used, seeded with seed. If seed is already a RandomState or Generator instance, then that object is used. Default is None.


This class is closely related to special_ortho_group.

Some care is taken to avoid numerical error, as per the paper by Mezzadri.



F. Mezzadri, “How to generate random matrices from the classical compact groups”, arXiv:math-ph/0609050v2.


>>> import numpy as np
>>> from scipy.stats import ortho_group
>>> x = ortho_group.rvs(3)
>>>, x.T)
array([[  1.00000000e+00,   1.13231364e-17,  -2.86852790e-16],
       [  1.13231364e-17,   1.00000000e+00,  -1.46845020e-16],
       [ -2.86852790e-16,  -1.46845020e-16,   1.00000000e+00]])
>>> import scipy.linalg
>>> np.fabs(scipy.linalg.det(x))

This generates one random matrix from O(3). It is orthogonal and has a determinant of +1 or -1.

Alternatively, the object may be called (as a function) to fix the dim parameter, returning a “frozen” ortho_group random variable:

>>> rv = ortho_group(5)
>>> # Frozen object with the same methods but holding the
>>> # dimension parameter fixed.


rvs(dim=None, size=1, random_state=None)

Draw random samples from O(N).