Linearization of a dynamical system

Linear instability is growth of small disturbances in the linearized dynamics. An eigenmode with positive temporal growth rate is sufficient to establish it. In a non-normal system, finite transient amplification is a different issue from the existence of an exponentially growing eigenmode. For normal modes written as $e^{i\alpha (x-ct)}$, the temporal rate is $\alpha \mathrm{Im}c$.