The covariance matrix of random variables is an matrix that . The correlation matrix is the covariance matrix of standardized random variables .
For any random variable , we have , where .
So the covariance matrix is positive-semidefinite, since the variance is non-negative.
Given correlation matrix, we can generate the random variables using Cholesky decomposition . Generate a vector of uncorrelated samples , then we have .
Another way is to use eigenvalue method. .