In a matrix model of the linearised force operator (the analogue of )

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Fluid and magnetohydrodynamic description · Fluid and magnetohydrodynamic description

Plasma Physics, Ch Whether it stays held: the energy principle, interchange, kink and tearing

Problem

In a matrix model of the linearised force operator (the analogue of −ω2ρ0ξ=F(ξ)-\omega^2\rho_0\boldsymbol\xi = \mathbf F(\boldsymbol\xi)), a stability calculation reduces to the real symmetric matrix A=(311−2),\mathsf A = \begin{pmatrix}3 & 1\\ 1 & -2\end{pmatrix}, whose Rayleigh quotient R(x)=xTAx/xTx\mathcal R(\mathbf x) = \mathbf x^{\mathsf T}\mathsf A\mathbf x/\mathbf x^{\mathsf T}\mathbf x equals ω2\omega^2 at each eigenvector of A\mathsf A and bounds its lowest eigenvalue from above for every trial vector x\mathbf x. ((a)) Evaluate R(x)\mathcal R(\mathbf x) for the trial vector x=(1,−3)\mathbf x = (1, -3). ((b)) State, with reason, whether this trial proves the presence of an unstable (growing) mode. ((c)) Find the exact eigenvalues of A\mathsf A and check the bound found in part (a) against the lowest one.

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