Theory and applications of the fission matrix method for continuous-energy Monte Carlo

•The fission matrix method is implemented into the MCNP Monte Carlo code.•Eigenfunctions and eigenvalues of power distributions are shown and studied.•Source convergence acceleration is demonstrated for a fuel storage vault problem.•Forward flux eigenmodes and relative uncertainties are shown for a reactor problem.•Eigenmodes expansions are performed during source convergence for a reactor problem.