  The authors have recently defined the R\'enyi information dimension rate
$d(\{X_t\})$ of a stationary stochastic process $\{X_t,\,t\in\mathbb{Z}\}$ as
the entropy rate of the uniformly-quantized process divided by minus the
logarithm of the quantizer step size $1/m$ in the limit as $m\to\infty$ (B.
Geiger and T. Koch, "On the information dimension rate of stochastic
processes," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Aachen, Germany, June
2017). For Gaussian processes with a given spectral distribution function
$F_X$, they showed that the information dimension rate equals the Lebesgue
measure of the set of harmonics where the derivative of $F_X$ is positive. This
paper extends this result to multivariate Gaussian processes with a given
matrix-valued spectral distribution function $F_{\mathbf{X}}$. It is
demonstrated that the information dimension rate equals the average rank of the
derivative of $F_{\mathbf{X}}$. As a side result, it is shown that the scale
and translation invariance of information dimension carries over from random
variables to stochastic processes.
