Suppose we have a variance covariance matrix $\Sigma$.
Under what conditions on the variance covariance matrix, $\Sigma$ is positive definite,
that is $\forall w \neq 0, w^T \Sigma w>0$.
In fact, I am considering the Markowitz model and want to ensure that the variance covariance matrix of the asset returns is positive definite.
Best Answer
If $\Sigma=E(XX^T)$, then you have $w^T\Sigma w=E((w^TX)^2)\geq 0$. That is why $\Sigma$ is always semi definite. It is not positive definite iff the components of the random vector $X$ are linearly dependent a.s.