Confused about Cholesky and eigen decomposition

I'm looking to generate correlated random variables. I have a symmetric, positive definite matrix. So I know that you can use the Cholesky decomposition, however I keep being told that this only works for Gaussian random variables?! Is that true?

Furthermore how does this compare to Eigen decomposition. For example using Cholesky decomposition we can write a random parameter as:

$x = \bar{x} + Lz$

where $L$ is the Cholesky decomposition (lower/upper triangular matrix) and $z$ is some vector of random variables. So one can sample the $z$'s and build up a pdf of x. Now we could also use Eigen decomposition and write x as:

$x = \bar{x} + U\lambda^{1\over2}z$

where $\lambda$ is a diagonal matrix of eigenvalues and $U$ is a matrix composed of the eigenvalues. So we could also build a pdf of this. But if we equate these $x$'s we find that $L = U\lambda^{1\over2}$ But this isn’t true as $L$ is triangular and $U\lambda^{1\over2}$ is not?! So I'm really, really confused. So to clarify the questions:

1) For Cholesky decomposition does the vector z have to be only Gaussian? 2) How does the eigenvalue compare with the Cholesky decomposition? They are clearly different factorisation techniques. So I don't see how the $x$'s above can be equivalent?

Thanks, as always, guys.

• (1) No, any distribution. (2) Hmm, I'm not mathematician. As I know the two decompositions are indeed quite different. Composed of the eigenvalues Did you mean "eigenvectors"? BTW, it is possible to generate the variables you want also via principal components, with the help of eigenvalues and eigenvectors. But, sorry, I failed to understand what you imply by the two formulas you showed. – ttnphns Jan 30 '14 at 17:20
• Take a look at this blog post. Hope this helps. – Aleksandr Blekh Dec 17 '14 at 2:23
• It's a theorem that if $x$ has a multivariate normal distribution, and we multiply $y=Ax$, then $y$ will also have a multivariate normal distribution. However, if $x$ has some other distribution then the distribution of $y$ need not be of the same family as the distribution of $x$. However, the method that you've described will ensure that you get the desired covariance. – Brian Borchers Feb 22 '15 at 22:12
• How about reading my answer at Stack Overflow: Obtain vertices of the ellipse on an ellipse covariance plot (created by car::ellipse). Although the question is asked in different application, the theory behind is the same. You will see nice figures for geometric explanation there. – 李哲源 Nov 4 '16 at 11:22

1) Pretty much yes. The reason is that the $x_i$'s are going to end up being a linear combination of the $z_i$'s. That works out nicely for Gaussian deviates because any linear combination of Gaussian deviates is, itself, a Gaussian deviate. Unfortunately, this is not necessarily true of other distributions.
2) It's a little puzzling, I know, but they are equivalent. Let $\Sigma$ be your covariance matrix and suppose you have both the Cholesky factorization, $\Sigma=L L^T$ and the eigendecomposition, $\Sigma=U \lambda U^T$. The covariance of $L z$ is given by: $$\begin{array}{} E[L z (L z)^T] & = & E[L z z^T L^T] \\ & = & L \ E[z z^T] \ L^T \\ & = & L \ I \ L^T \\ & = & L L^T \\ & = & \Sigma \end{array}$$ Similarly, the covariance of $U \lambda^\frac{1}{2} z$ is given by: $$\begin{array}{} E[U \lambda^\frac{1}{2} z (U \lambda^\frac{1}{2} z)^T] & = & E[U \lambda^\frac{1}{2} z z^T \lambda^\frac{1}{2} U^T] \\ & = & U \lambda^\frac{1}{2} \ E[z z^T] \ \lambda^\frac{1}{2} U^T \\ & = & U \lambda^\frac{1}{2} \ I \ \lambda^\frac{1}{2} U^T \\ & = & U \lambda^\frac{1}{2} \lambda^\frac{1}{2} U^T \\ & = & U \lambda U^T \\ & = & \Sigma \end{array}$$ For purposes of computation, I suggest you stick with the Cholesky factorization unless your covariance matrix is ill-conditioned/nearly singular/has a high condition number. Then it's probably best to switch to the eigendecomposition.
• Oh, I should add that this technique works for ANY factorization of $\Sigma = F F^T$. And this factorization is not unique unless you force $F$ to be lower triangular (in which case, $F$ is the Cholesky factor). – Bill Woessner Oct 30 '15 at 14:54