function M=RandOrthMat(n, tol)
% M = RANDORTHMAT(n)
% generates a random n x n orthogonal real matrix.
%
% M = RANDORTHMAT(n,tol)
% explicitly specifies a thresh value that measures linear dependence
% of a newly formed column with the existing columns. Defaults to 1e-6.
%
% In this version the generated matrix distribution *is* uniform over the manifold
% O(n) w.r.t. the induced R^(n^2) Lebesgue measure, at a slight computational
% overhead (randn + normalization, as opposed to rand ).
%
% (c) Ofek Shilon , 2006.
if nargin==1
tol=1e-6;
end
M = zeros(n); % prealloc
% gram-schmidt on random column vectors
vi = randn(n,1);
% the n-dimensional normal distribution has spherical symmetry, which implies
% that after normalization the drawn vectors would be uniformly distributed on the
% n-dimensional unit sphere.
M(:,1) = vi ./ norm(vi);
for i=2:n
nrm = 0;
while nrm