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. .

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