<p>In her seminal 1989 work, Greenbaum demonstrated that the Jacobi matrices produced by the finite precision Lanczos algorithm after <i>k</i> iterations can be interpreted as the results of the exact Lanczos algorithm applied to a larger matrix, whose eigenvalues lie in small intervals around those of the original matrix. This establishes a mathematical model for finite precision Lanczos computations. The present work extends some of these ideas to the block Lanczos algorithm. A generalization of the continuation process is proposed and shown to terminate in a finite number of iterations using carefully constructed perturbations. By deriving sufficient conditions that keep the required perturbations small, it is shown that the eigenvalues of the model matrix stay close to those of the original matrix. While in the single-vector case these conditions are always satisfiable, the question of whether they can always be satisfied in the block case remains open. Finally, numerical experiments demonstrate a practical implementation of the continuation process, empirically assess the validity of the sufficient conditions, and plot the sizes of the perturbations.</p>