A linearised dynamical model that cannot be diagonalised is represented by the Jordan…

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

Error, Chance and the Machine, Ch Rounding, backward error and the sensitivity of a computation

Problem

A linearised dynamical model that cannot be diagonalised is represented by the 3×33\times3 Jordan block J=(510051005),\mathsf J = \begin{pmatrix}5 & 1 & 0\\ 0 & 5 & 1\\ 0 & 0 & 5\end{pmatrix}, with the single eigenvalue λ=5\lambda = 5 repeated three times. A small correction to the model changes the bottom-left entry from 00 to ε=10−6\varepsilon = 10^{-6}: Jε=(510051ε05).\mathsf J_\varepsilon = \begin{pmatrix}5 & 1 & 0\\ 0 & 5 & 1\\ \varepsilon & 0 & 5\end{pmatrix}. ((a)) Writing μ=5−x\mu = 5 - x, expand det⁡(Jε−xI)\det(\mathsf J_\varepsilon - x\mathsf I) and show that the eigenvalues of Jε\mathsf J_\varepsilon satisfy μ3+ε=0\mu^3 + \varepsilon = 0. ((b)) Solve for the three values of μ\mu, and hence the three eigenvalues of Jε\mathsf J_\varepsilon, for ε=10−6\varepsilon = 10^{-6}. ((c)) State the general scaling with ε\varepsilon this reveals for an n×nn\times n Jordan block perturbed in one corner, and contrast it with the exact Weyl bound for a symmetric matrix perturbed by an entry of the same size ε\varepsilon.

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