A gyroscopic coupling between two coordinates is represented by A = pmatrix 0&2 -2&0…

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

Conditions of Validity, Ch The spectral theorem, degeneracy and the matrices that will not diagonalise

Problem

A gyroscopic coupling between two coordinates is represented by A=(02−20).\mathsf{A} = \begin{pmatrix}0&2\\-2&0\end{pmatrix}. ((a)) Someone argues: "only symmetric matrices are diagonalised by an orthonormal eigenbasis, and A\mathsf{A} is not symmetric (AT=−A\mathsf{A}^{\mathrm{T}} = -\mathsf{A}), so no orthonormal eigenbasis exists for A\mathsf{A}." Find the error, and give the orthonormal (complex) eigenbasis A\mathsf{A} does have. ((b)) A mounting bracket has potential-energy Hessian H=(5445) N m−1\mathsf{H} = \begin{pmatrix}5&4\\4&5\end{pmatrix}\,\mathrm{N\,m^{-1}} at an equilibrium point. Someone argues: "the off-diagonal coupling, 4 N m−14\,\mathrm{N\,m^{-1}}, is nearly as large as the diagonal entries, 5 N m−15\,\mathrm{N\,m^{-1}}, so this must be a saddle, as for the potential V=x2+4xy+y2V=x^2+4xy+y^2 earlier in the chapter." Find the error and classify the equilibrium correctly.

Browse the problem library