The integral transport equation for the flux density in the interior of an infinite homogeneous cylinder is reduced to a matrix eigenvalue problem for the critical cylinder and a set of linear algebraic equations for the driven case with surface in-currents. The matrix elements are identified as moments of modified Bessel functions and are easily computed. A connection is made with classical diffusion theory via a related matrix eigenvalue problem, from which the diffusion coefficient and extrapolation endpoint can be computed. Analytic properties of the matrix elements are used to obtain approximate solutions for (optically) dense and transparent cylinders. Numerical results are given for the American Nuclear Society benchmark black rod problem, and the fact that only small matrices are required for a large range of problems is demonstrated.