Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 78 / 5 / a / Solution

Use the Euclidean norm and its induced operator norm. The spectral theorem for real symmetric matrices gives $\Vert A\Vert ={\lambda }_n$ and $\Vert A^{-1}\Vert =1/{\lambda }_1$. If $e=y_{\delta }-y$, then the direct inverse has error $A^{-1}e$, soFor nonzero $y$, the relation $\Vert y\Vert =\Vert Ax\Vert \le {\lambda }_n\Vert x\Vert$ also givesThis is the spectral condition number of a positive-definite matrix. The worst relative amplification is attained by exact data along an eigenvector for ${\lambda }_n$ and noise along one for ${\lambda }_1$. A large $\kappa$ therefore makes the reconstruction sensitive to relative data errors. Absolute sensitivity additionally depends on ${\lambda }_1$ itself: scaling every eigenvalue equally leaves $\kappa$ unchanged but changes $1/{\lambda }_1$. Since the matrix is finite dimensional and positive definite, the inverse exists and is continuous; the issue is ill-conditioning rather than mathematical nonexistence or discontinuity.