Past exam of the numerical linear algebra course of the University of Oxford / 2025 / 1 / b / iv

No uniform absolute error bound of order $ϵ$ follows. Take the Jordan blockHere $\Vert B{\Vert }_2=1$ and $\Vert E{\Vert }_2=ϵ$. The exact eigenvalues of $B$ are both zero, whereasThese are admissible outputs under a backward error guarantee of size $ϵ$, yet their forward errors are $\sqrt{ϵ}$. This is the square-root splitting of a defective double eigenvalue. More generally, eigenvalue sensitivity depends on the matrix; backward stability alone does not control it. The singular value perturbation bound used in part (ii) has no equally uniform counterpart for the eigenvalues of arbitrary nonsymmetric matrices.