  We develop a new bisimulation (pseudo)metric for weighted finite automata
(WFA) that generalizes Boreale's linear bisimulation relation. Our metrics are
induced by seminorms on the state space of WFA. Our development is based on
spectral properties of sets of linear operators. In particular, the joint
spectral radius of the transition matrices of WFA plays a central role. We also
study continuity properties of the bisimulation pseudometric, establish an
undecidability result for computing the metric, and give a preliminary account
of applications to spectral learning of weighted automata.
