The question is ambiguous. Are you talking about the covariance between two matrices of random elements, or the covariance arising from two sets of linear transformations of the same random vectors? In any case, it will involve the second order moments. For two random matrices, the covariance would be the matrix of covariances of their Kronecker, or direct product. In the case of linear transformations, the covariance depends on the joint distribution of the random vectors as well as the implied linear relationships.
A zero covariance implies that two variables are "uncorrelated". It does not imply that they are necessarily independent.
