Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 331 / 2 / a / Solution

The energy isThusThis quadratic form can be negative for positive $c_1,c_2$ precisely whenThe decaying vectors must therefore be sufficiently nonorthogonal and oppositely directed. This is transient growth from non-normal modes.

With $c_2=1$ and $\mu =c_1/c_2$, maximizing $E^{\prime }(0)$ givesso