We associate a dynamical system with input-output networks and study its spectral properties. Specifically, we develop a dynamic production network model featuring adjustment costs of changing inputs and thus gradual recovery from temporary TFP shocks. First, we explicitly solve for the output and welfare effects of temporary shocks. We show shocks to sectors that generate significant sales through distant linkages to the consumer are most damaging. Second, we eigendecompose the input-output matrix and show, because higher-order linkages take longer to recover, fewer eigenvectors are needed to represent the welfare impact of sectoral shocks in the dynamic economy compared to the Domar weights. Third, we analyze the U.S. input-output structure and show the welfare impact of temporary shocks has a low-dimensional, 4-factor structure (out of 171 eigenvectors). Finally, we revisit the historical use of input-output analysis in target selection for bombing Nazi Germany and Imperial Japan during WWII.