The vector μ in these conditions is the expected value of X and the matrix Γ=A^{T}A is the covariance matrix of the components X_{i}. It is important to realize that the covariance matrix must be allowed to be singular. That case arises frequently in statistics; for example, in the distribution of the vector of residuals in ordinary linear regression problems. Note also that the X_{i} are in general not independent; they can be seen as the result of applying the linear transformation A to a collection of independent Gaussian variables Z.
Proof?
Multivariate Gaussian density
Recall characteristic function of a random vector.
Recall characterizations of gaussian random variables.
Calculate characteristic function of Z in terms of characteristic function of X.
Deduce characteristic functional of X in terms of mean vector and covariance matrix.
Search Encyclopedia

Featured Article
