A system of degrees of freedom near a stable equilibrium has real symmetric matrices…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Linear algebra and vector spaces

Action and Symmetry, Ch Normal modes as a generalised eigenvalue problem

Problem

A system of nn degrees of freedom near a stable equilibrium has real symmetric matrices T\mathbf T and V\mathbf V (so TT=T\mathbf T^{\mathrm T}=\mathbf T, VT=V\mathbf V^{\mathrm T}=\mathbf V), with T\mathbf T positive definite: ξTTξ>0\boldsymbol\xi^{\mathrm T}\mathbf T\boldsymbol\xi>0 for every real ξ≠0\boldsymbol\xi\ne\mathbf 0. A normal mode satisfies (V−ω2T)a=0(\mathbf V-\omega^2\mathbf T)\mathbf a=\mathbf 0 for some scalar ω2\omega^2, a priori possibly complex, and some vector a≠0\mathbf a\ne\mathbf 0, also a priori possibly complex. ((a)) Prove that every root ω2\omega^2 of det⁡(V−ω2T)=0\det(\mathbf V-\omega^2\mathbf T)=0 is in fact real. ((b)) State exactly where the symmetry of V\mathbf V and T\mathbf T, and the positive definiteness of T\mathbf T, are each used in the proof.

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