# Does one use the covariance or correlation matrix in cholesky decomposition to generate correlated samples

Can we interchangeably use Cholesky decomposition of covariance and correlation matrix to generate simulations? If not, in which situations do we use one or the other and why? Thanks in advance.

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The usual approach is to decompose the correlation matrix. See e.g. here sitmo.com/article/generating-correlated-random-numbers Given the relationship between correlation and covariance $\rho_{XY}=\frac{cov(X,Y)}{\sigma_X \sigma_Y}$ you can always turn a covariance matrix into the correlation one. –  Probilitator Mar 23 '14 at 9:25

You can use the either, as both necessarily are symmetric positive definite; covariance is a personal preference. It's really just a matter of scaling, as $\mathcal{N}(0,\Sigma)$ is distributionally $\sqrt{\Sigma} \mathcal{N}(0,1)$.
Correlation would require additional scaling (i.e. multiplication of every $\mathcal{N}(0,\rho)$ element by its respective volatility, and therefore requires more operations).