Mutual information between two Gaussian distribution

Question

Suppose we have two variables $x_i$ and $x_j$ with covariance matrices $P_i$ and $P_j$ and cross-covariance $P_{ij}$. I'd like to find the mutual information on them. From reverse engineering of some given equations on fusing two variable $x_i$ and $x_j$, I found that it should be $x_m=P_{ij}(P_i)^{-1} x_i$ with some covariance matrix $P_m$, which has analogy with the definition of projection in vector space. But I do not know how to obtain it directly and can not find any reference on it. I would appreciate your comments and helps.