A scalarization technique for computing the power and exponential moments of Gaussian random matrices

We consider the problems of computing the power and exponential moments EX^s and Ee^tX of square Gaussian random matrices X = A + BWC for positive integer s and real t, where W is a standard normal random vector and A, B, C are appropriately dimensioned constant matrices. We solve the problems by a matrix product scalarization technique and interpret the solutions in system-theoretic terms. The results of the paper are applicable to Bayesian prediction in multivariate autoregressive time series and mean-reverting diffusion processes.