A calibration rig holds a test mass with two uncoupled sensor springs along…

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

Mathematical Methods for Physics, Ch Floating-point arithmetic, conditioning and numerical stability

Problem

A calibration rig holds a test mass with two uncoupled sensor springs along perpendicular directions, of equal stiffness. Their combined stiffness matrix is A=(2.0×10−4002.0×10−4) N m−1.\mathsf{A} = \begin{pmatrix}2.0\times10^{-4} & 0\\ 0 & 2.0\times10^{-4}\end{pmatrix}\,\mathrm{N\,m^{-1}}. Its determinant is det⁡A=4.0×10−8 N2 m−2\det\mathsf{A} = 4.0\times10^{-8}\,\mathrm{N^2\,m^{-2}}, which looks very small. Someone argues: "Because the determinant is so small, this matrix must be severely ill-conditioned, and any displacement computed from a force by solving Ax=F\mathsf{A}\mathbf{x} = \mathbf{F} will lose most of its precision." ((a)) Compute the condition number of A\mathsf{A} from its eigenvalues. ((b)) Say whether the argument is correct, and explain what the small determinant is actually telling us.

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