= Solution
\b[No uniform absolute error bound of order $\epsilon$ follows.] Take the <Jordan block>
$$
B=\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad
E=\begin{pmatrix}0&0\\\epsilon&0\end{pmatrix}.
$$
Here $\|B\|_2=1$ and $\|E\|_2=\epsilon$. The exact <eigenvalues> of $B$ are both zero, whereas
$$
\det(\lambda I_2-B-E)=\lambda^2-\epsilon,
\qquad \boxed{\lambda_\pm(B+E)=\pm\sqrt\epsilon}.
$$
These are admissible outputs under a <backward error> guarantee of size $\epsilon$, yet their <forward errors> are $\sqrt\epsilon$. 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>.
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