In this paper, eigenvalue sensitivity measures are proposed that are suitable for assessing the fragility of digital controllers and filters implemented using floating-point arithmetic. Floating-point arithmetic parameter uncertainty is shown to be multiplicative. Based on first-order eigenvalue sensitivity analysis, an upper bound on the eigenvalue perturbations is derived. Consequently, open-loop and closed-loop eigenvalue sensitivity measures are proposed. These measures are dependent upon the filter or controller realization. Problems of obtaining the optimal realization with respect to both the open-loop and the closed-loop eigenvalue sensitivity measures are posed. The problem for the open-loop case is completely solved. The problem for the closed-loop case is solved using nonlinear programming. The problems are illustrated with a numerical example.