  We study the properties of the eigenvalues of real random matrices and their
products. It is known that when the matrix elements are Gaussian-distributed
independent random variables, the fraction of real eigenvalues tends to unity
as the number of matrices in the product increases. Here we present numerical
evidence that this phenomenon is robust with respect to the probability
distribution of matrix elements, and is therefore a general property that
merits detailed investigation. Since the elements of the product matrix are no
longer distributed as those of the single matrix nor they remain independent
random variables, we study the role of these two factors in detail. We study
numerically the properties of the Hadamard (or Schur) product of matrices and
also the product of matrices whose entries are independent but have the same
marginal distribution as that of normal products of matrices, and find that
under repeated multiplication, the probability of all eigenvalues to be real
increases in both cases, but saturates to a constant below unity showing that
the correlations amongst the matrix elements are responsible for the approach
to one. To investigate the role of the non-normal nature of the probability
distributions, we present a thorough analytical treatment of the $2 \times 2$
single matrix for several standard distributions. Within the class of smooth
distributions with zero mean and finite variance, our results indicate that the
Gaussian distribution has the maximum probability of real eigenvalues, but the
Cauchy distribution characterised by infinite variance is found to have a
larger probability of real eigenvalues than the normal. We also find that for
the two-dimensional single matrices, the probability of real eigenvalues lies
in the range [5/8,7/8].
