Linearization

Linearization replaces a nonlinear relation near a reference state by its first-order approximation.

Working to linear order in a small parameter retains constant and first-degree terms and discards all terms of quadratic and higher order.

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 $\alpha \mathrm{Im}c$.