Three identical sensors are coupled symmetrically so that their coupling matrix is A…

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

Three identical sensors are coupled symmetrically so that their coupling matrix is A=(111111111).\mathsf{A} = \begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}. Its eigenvalues are 3,0,03, 0, 0. ((a)) Someone argues: "the eigenvalue 00 is repeated, so A\mathsf{A} cannot have three independent eigenvectors and cannot be diagonalised." Find the error, and exhibit an orthonormal eigenbasis for A\mathsf{A}. ((b)) At a stationary point of a potential, the Hessian is H=(400−1) N m−1\mathsf{H} = \begin{pmatrix}4&0\\0&-1\end{pmatrix}\,\mathrm{N\,m^{-1}}. Someone argues: "the trace, 4+(−1)=3 N m−14 + (-1) = 3\,\mathrm{N\,m^{-1}}, is positive, so this is a minimum." Find the error and classify the stationary point correctly.

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