Generating correlated random numbers: Why does Cholesky decomposition work? Let's say I want to generate correlated random variables. I understand that I can use Cholesky decomposition of the correlation matrix to obtain the correlated values. If $C$ is the correlation matrix, then we can do the cholesky decomposition:
$LL^{T}=C$
Then I can easily generate correlated random variables:
$LX=Y$,
where $X$ are uncorrelated values and $Y$ are correlated values. If I want two correlated random variables then $L$ is:
$L = \left[ {\begin{array}{*{20}c}
   1 & 0  \\
   \rho  & {\sqrt {1 - \rho ^2 } }  \\
\end{array}} \right]
$
I understand that this works, but I don't really understand why... My question is: Why does this work?
 A: The co-variance matrix of any random vector $Y$ is given as $\mathbb{E} \left(YY^T \right)$, where $Y$ is a random column vector of size $n \times 1$. Now take a random vector, $X$, consisting of uncorrelated random variables with each random variable, $X_i$, having zero mean and unit variance $1$. Since $X_i$'s are uncorrelated random variables with zero mean and unit variance, we have $\mathbb{E} \left(X_iX_j\right) = \delta_{ij}$. Hence, $$\mathbb{E} \left( X X^T \right) = I$$ To generate a random vector with a given covariance matrix $Q$, look at the Cholesky decomposition of $Q$ i.e. $Q = LL^T$. Note that it is possible to obtain a Cholesky decomposition of $Q$ since by definition the co-variance matrix $Q$ is symmetric and positive definite.
Now look at the random vector $Z = LX$. We have $$\mathbb{E} \left(ZZ^T\right) = \mathbb{E} \left((LX)(LX)^T \right) = \underbrace{\mathbb{E} \left(LX X^T L^T\right) = L \mathbb{E} \left(XX^T \right) L^T}_{\text{ Since expectation is a linear operator}} = LIL^T = LL^T = Q$$ Hence, the random vector $Z$ has the desired co-variance matrix, $Q$.
A: The specified correlation, C, can have negative element, i.e. Negative correlation. But, C, must Be positive definite to be a proper corr. matrix. E.g. [ 1 -0.9 0.5
  -0.9 1 -0.1
  0.5 -0.1 1 ]
Is ok, but 0.9,-0.5,0.1 (in all places) produces a negative definite matrix, so not ok.
Actually, for all values positive will have the effect that the sign of the second random variable changes.
