One usually defines the space $$\{M\in\mathrm{Mat}_n(\mathbb R):\forall j, m_{jj}=1,M^\top=M\ge0\}$$ to be the space of correlation matrices. It is clear that every correlation matrix is inside this space. But how to show that for each $M$ in this space it associates with a random vector $X$ whose correlation matrix is exactly $M$, i.e. how to construct such $X$?
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actually a correlation matrix is positive-semidefinite – phaedo Oct 30 '18 at 01:46
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I do not think you can take any random vector X, but if X is i.i.d. standard normal then a common numerical technique to obtain correlated standard normals is to multiply the matrix $\mathbf u$ of samples from X by the Cholesky decomposition of M
I believe as the sample size $n\to\infty$ the resulting correlation matrix converges to M
phaedo
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1Thanks. Based on your answer I found https://math.stackexchange.com/questions/446093/generate-correlated-normal-random-variables – JJJZZZZZ Oct 30 '18 at 05:00
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Actually I think that chol(M)*X will have exact correlation matrix M (no need to use samples) – phaedo Oct 30 '18 at 13:51