Constructing large covariance matrix from correlations and covariances gives non positive definite matrix.

Let's create a large correlation matrix (simply a symmetric matrix with ones on the diagonal and values between -1 and 1 on the off-diagonals), and a vector of standard deviations (positive reals).

dim = 400;
corr_ = eye(dim);
for ii = 1:dim
    stdev_ = rand;
    for jj = ii+1:dim
        corr_(ii,jj) = rand*2-1;
        corr_(jj,ii) = corr_(ii,jj);
    end
end

Now lets create the covariance matrix in two different ways.

covmat1 = (stdev_*stdev_)'.*corr_;
covmat2 = diag(stdev_)*corr_*diag(stdev_);

Why are both of them not positive definite?

chol(covmat1)
chol(covmat2)

[Jeff Miller]

Not sure of the answer to your final question, but it may be relevant that your method of generating a correlation matrix is not valid (i.e., it generates a lot of matrices that could not possibly be correlation matrices). The different correlation values within a matrix are highly constrained--for example, if corr(1,2) = 0.99 and corr(1,3)=0.99 then corr(2,3) has to be quite large as well. So, you can't just generate correlation values independently.

[Michael Stollenwerk]

Yes, thank you very much! That makes sense. I guess in the data I am provided (a 400 by 400 correlation matrix) some entries must be slightly off s.th. it isn't a correlation matrix.