Is the matrix representation of a multivariate normal unique? Under this definition of multivariate normal,

Is the matrix and mean tuple $(D, \mu)$ unique? Any proof?

The vector $\mu$ is unique, since it must just be the expected value of $X$ (since $E(DW)=DE(W)=0$ no matter what $D$ is).  However, $D$ is not unique (unless the dimension is $0$).  The simplest way to see this is to consider when the distribution of $X$ is already standard normal.  Then $-X$ is also standard normal, so $D$ could be either the identity matrix $I$ or $-I$.  More generally, $D$ could be any orthogonal matrix, since the multivariate standard normal distribution is invariant under orthogonal transformations.
