Past exam of the Probability, statistics and options II course of the University of Oxford / 2007 / P / 4 / f

The claimed endpoint bounds with radius $ϵ$ are false for positive corner perturbations. For example, take $n=3$, $ϵ=1/1000$ and $\delta =9/10<1-ϵ$. The characteristic polynomial isThroughout $|\lambda -1|\le 1/1000$, the absolute value of the sum of its first two terms is below $0.003$, whereas $9/160=0.05625$. Hence this interval contains no root, contradicting the first requested bound. The actual eigenvalues are approximately $1.029144,1.943599,3.027257$, which also violate the last requested bound.

There is a valid real-eigenvalue conclusion with wider endpoint bounds. For $0<ϵ<1/4$ and $\delta <(n-1)/2-ϵ$, the Gershgorin discs from rows have one enlarged component on the left; the Gershgorin discs from columns have one enlarged component on the right. Each has an enlarged radius $r=\delta +ϵ<(n-1)/2$. A small disc that meets the enlarged left disc has index $j$ satisfying $j-1\le r+q<n/2$. A small disc meeting the enlarged right disc satisfies $n-j<n/2$. Thus, when $n=2m$, every disc with index at most $m$ is isolated in the column family and every disc with index at least $m+1$ is isolated in the row family. Their counts give all $n$ eigenvalues, one in each disjoint small disc, so all are real.

When $n=2m+1$, the same argument isolates the first $m$ discs in the column family and the last $m$ in the row family. It accounts for $2m$ real eigenvalues. The one remaining eigenvalue is real by conjugation, and belongs to the central small disc: the enlarged left and right discs are disjoint, and their two components can intersect only within the disc centered at $m+1$. Therefore the valid uniform bounds areThe sharper endpoint radii $ϵ$ hold in part (d) when $\delta =0$, and cannot simply be carried over to the perturbed matrix.