  We define two a priori tests of pseudo-random number generators for the class
of linear matrix-recursions. The first desirable property of a random number
generator is the smallness of serial or lagged correlations between generated
numbers. For the particular matrix generator called MIXMAX, we find that the
serial correlation actually vanishes. Next, we define a more sophisticated
measure of correlation, which is a multiple correlator between elements of the
generated vectors. The lowest order non-vanishing correlator is a four-element
correlator and is non-zero for lag $s=1$. At lag $s \ge 2$, this correlator
again vanishes. For lag $s=2$, the lowest non-zero correlator is a six-element
correlator. The second desirable property for a linear generator is the
favorable structure of the lattice which typically appears in dimensions higher
than the dimension of the phase space of the generator, as discovered by
Marsaglia. We define an appropriate generalization of the notion of the
spectral index for LCG which is a measure of goodness of this lattice to the
matrix generators such as MIXMAX and find that the spectral index is
independent of the size of the matrix N and is equal to $\sqrt{3}$.
