  We discuss the product of independent induced quaternion ($\beta=4$) Ginibre
matrices, and the eigenvalue correlations of this product matrix. The joint
probability density function for the eigenvalues of the product matrix is shown
to be identical to that of a single Ginibre matrix, but with a more complicated
weight function. We find the skew-orthogonal polynomials corresponding to the
weight function of the product matrix, and use the method of skew-orthogonal
polynomials to compute the eigenvalue correlation functions for product
matrices of finite size. The radial behavior of the density of states is
studied in the limit of large matrices, and the macroscopic density is
discussed. The microscopic limit at the origin, at the edge(s) and in the bulk
is also discussed for the radial behavior of the density of states.
