A coupling matrix between two channels is A = pmatrix 2&3 1&4 pmatrix

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 coupling matrix between two channels is A=(2314).\mathsf{A} = \begin{pmatrix}2&3\\1&4\end{pmatrix}. Its eigenvalues are found from det⁡(A−λI)=0\det(\mathsf{A}-\lambda\mathsf{I})=0 to be λ=5\lambda=5 and λ=1\lambda=1. ((a)) Someone argues: "the eigenvalues are real, so by the reality and orthogonality theorem the eigenvectors of A\mathsf{A} must be orthogonal." Find the error. ((b)) A stationary point has Hessian H=(−300−6) N m−1\mathsf{H} = \begin{pmatrix}-3&0\\0&-6\end{pmatrix}\,\mathrm{N\,m^{-1}}. Someone argues: "the determinant, 18 N2 m−218\,\mathrm{N^2\,m^{-2}}, is positive, so this is a minimum." Find the error and classify the point correctly.

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