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I've read that decomposition of a correlation matrix (Cholesky or Spectral) is only valid if all of the random variables have a normal distribution. Is this correct? If so, how do I correlate random numbers chosen from different probability distributions?

There is no relation between normality and matrix decomposition. The job of choleski decomposition is to find "square root" of a PD matrix. As the square root of a positive number not unique, "square root" of a matrix similarly not unique. You can use SVD etc to get the same, all are correct. But Choleski procedure is mostly adopted bcoz it requires least calculation. As upper diagonal (or lower diagonal) elements are all 0.Regards,

Thanks.

Thanks.

in principle you can use choleski, however for non normal distributions correlation can be non sensical for comovement between rvs

Thanks. Yes, I have just read Chapter 5 of Peter Jackel's book "Monte Carlo Methods in Finance" and it's clear to me now. I was confusing the two separate issues of Cholesky Decomposition and Linear Correlation. It's the matrix of linear correlation coefficients that can lead to nonsense results if the RVs are not all normal.