Is the covariance matrix of multivariate normal distribution has to be positive definite?

Let $\Sigma$ be the covariance matrix of a multivariate normal distribution.

Thereotically, I guess it does not have to to strictly positive definition, i.e., $\det (\Sigma )=0$ is possible.

In this case, the density function does not exist, how can I generate random numbers(vectors) from such multivariate normal distribution?

• To simulate $X$ with covariance $\Sigma$ or size $n$ and rank $k$, choose $L$ of size $n\times k$ such that $\Sigma=LL^T$, simulate $U$ standard normal of size $k$ and use $X=LU$. – Did Sep 2 '17 at 19:25
• @MANMAID "so I am not sure, if this works or not" Or not. – Did Sep 2 '17 at 19:26