If is a Gaussian vector, an matrix and , then is also a Gaussian vector.

For any , let ; then

a Gaussian random variable plus a constant, hence a Gaussian random variable.

For a Gaussian vector define

The entries of are , so is a symmetric matrix, and it is non-negative definite since

Moreover for every , so

Since the MGF uniquely characterises the distribution when finite on an open set, a Gaussian vector is completely determined by and .

If are i.i.d. and , then is a Gaussian vector: for every ,

so ; we write .

Conversely, given any non-negative definite matrix with spectral decomposition and eigenvalues , the square root

lets us construct, for any mean ,

As a linear transformation of a Gaussian vector this is again a Gaussian vector, with

so exists for every non-negative definite covariance matrix.

If is positive definite, then has density

This follows from the change-of-variables formula applied to with of density .

If instead with , no density on exists: after an orthogonal change of basis one may assume

and then

so the distribution is supported on an affine subspace of dimension .

If are independent then for , so the covariance matrix of a Gaussian vector is diagonal in that case. For Gaussian vectors the converse holds: if is diagonal and strictly positive definite with diagonal , the joint density factorises as

so by the factorisation criterion the coordinates are independent with ; alternatively the MGF

factorises into functions of the individual , and uniqueness of the MGF again gives independence.

Hence, for a Gaussian vector, are independent precisely when for all .