It's not too hard to see that any correlation matrix must have certain properties, such as all entries in the range -1 to 1, symmetric, positive semi-definite (excluding pathological cases like singular matrices for the moment).
But I'm wondering about the other direction. If I write down some matrix that is positive semi-definite, is symmetric, has 1's along the main diagonal, and all entries are between -1 and 1, does this guarantee that there exists a set of random variables giving rise to that correlation matrix?
If the answer is easy to see, a helpful hint about how to define the set of random variables that would give rise to a given matrix would be appreciated.