= Solution
Self-adjointness gives $\langle\boldsymbol\xi,F\boldsymbol\xi\rangle=\langle F\boldsymbol\xi,\boldsymbol\xi\rangle=\langle\boldsymbol\xi,F\boldsymbol\xi\rangle^*$, so
$$
\boxed{W[\boldsymbol\xi]\in\mathbb R.}
$$
The integrated expression from part (a) also writes it explicitly as
$$
W[\boldsymbol\xi]=\frac12\int\left[\rho\xi_i^*\Phi_{,ij}\xi_j+(\partial_j\xi_i^*)V_{ijkl}(\partial_l\xi_k)\right]dV.
$$
This is the quadratic change in <potential energy> associated with the perturbation, including <pressure>, magnetic and imposed-gravity restoring effects. Its sign need not be positive term by term.
The perturbation <kinetic energy> is $T=\tfrac12\langle\dot{\boldsymbol\xi},\dot{\boldsymbol\xi}\rangle$, and the total quadratic perturbation <energy> is
$$
\boxed{E=T+W=\tfrac12\langle\dot{\boldsymbol\xi},\dot{\boldsymbol\xi}\rangle-\tfrac12\langle\boldsymbol\xi,F\boldsymbol\xi\rangle.}
$$
Since the equilibrium and $F$ are time independent, differentiating and using the <self-adjoint operator> identity gives
$$
\frac{dE}{dt}=\operatorname{Re}\langle\dot{\boldsymbol\xi},\ddot{\boldsymbol\xi}\rangle-\operatorname{Re}\langle\dot{\boldsymbol\xi},F\boldsymbol\xi\rangle=0.
$$
Thus $\boxed{dE/dt=0}$ along the linearized motion. This conserved kinetic-plus-restoring <energy> is the basis of the <magnetohydrodynamic energy principle>.
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