A stiffness matrix measured with lengths in millimetres and forces in newtons is A =…

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

Conditions of Validity, Ch The four subspaces and the Fredholm alternative

Problem

A stiffness matrix measured with lengths in millimetres and forces in newtons is A=(5225) N mm−1.\mathsf{A} = \begin{pmatrix} 5 & 2 \\ 2 & 5 \end{pmatrix}\ \mathrm{N\,mm^{-1}}. Converting to SI units (metres) multiplies every entry by 10310^3, giving B=103 A\mathsf{B} = 10^3\,\mathsf{A} in N m−1\mathrm{N\,m^{-1}}. ((a)) Find the eigenvalues of A\mathsf{A}, hence det⁡A\det\mathsf{A} and the condition number κ(A)\kappa(\mathsf{A}). ((b)) Find det⁡B\det\mathsf{B} and κ(B)\kappa(\mathsf{B}). ((c)) State the general rule relating det⁡(sA)\det(s\mathsf{A}) to det⁡A\det\mathsf{A} for a scalar ss and an n×nn\times n matrix, and explain why κ\kappa is unaffected by this rescaling while the determinant is not.

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