  In this note, we define a Gaussian probability distribution over matrices. We
prove some useful properties of this distribution, namely, the fact that
marginalization, conditioning, and affine transformations preserve the matrix
Gaussian distribution. We also derive useful results regarding the expected
value of certain quadratic forms based solely on covariances between matrices.
Previous definitions of matrix normal distributions are severely
under-parameterized, assuming unrealistic structure on the covariance (see
Section 2). We believe that our generalization is better equipped for use in
practice.

Probability