Can a multivariate Gaussian always be factorized into a product of univariate Gaussians?

Question

It can be shown that a multivariate Gaussian with diagonal covariance matrix can be factorized into a product of univariate Gaussians, which means that the variables are independent (with independent I mean the probabilistic concept, i.e. $p(x,y) = p(x)p(y)$).

However, if we have a multivariate Gaussian with a non-diagonal covariance matrix, we can switch into the basis of the eigenvectors of the covariance matrix to make it diagonal (whitening).

Is this always possible? Because, if so, we could always transform a multivariate Gaussian in such a way that its variables become independent.

Answer 1

This is always possible as long as the distribution is not degenerate. In other words, as long as the covariance matrix is nonsingular. This can be seen as a Gram-Schmidt procedure. In fact, this is how one simulates a multivariate random variable in computing software packages such as R and MATLAB. We start from a standard normal random vector, which is zero mean and identity covariance. Then we multiply the simulated standard normal by an appropriate matrix and shift with a constant to get a sample from a desired multivariate normal distribution.

Answer 2

The variance - covariance matrix is positive definite, so it can be diagonalized. As a corollary we get linear combinations of the variables which are independent normal variables.