Let $X$ be a random vector. It is well known that the covariance matrix $\Sigma$ of $X$ is a square matrix that has the following properties:
(1) It is symmetric;
(2) Its diagonal entries are nonnegative;
(3) It is positive semidefinite.
Now, does the converse also holds? In other words, suppose we have a square matrix $M$ which has properties (1) through (3). Can we say that there exists a random vector $X$ whose covariance matrix is equal to $M$?
More in general, which criteria do we know to show that some matrix $M$ is actually a covariance matrix?