Past exam of the options II course of the University of Oxford / 2007 / P / P4 / f / Solution

The endpoint bounds printed in this part are false. Take $n=3$, $ϵ=1/100$ and $\delta =9/10$, which obey $\delta <(n-1)/2-ϵ=99/100$. The characteristic polynomial isIt has $P(101/100)=-31599/10^6<0$ and $P(5/4)=441/1600>0$, so it has a real root strictly between $1.01$ and $1.25$, outside the first printed interval. There is a unique eigenvalue in the transpose's isolated first disc of radius $1/4$, identifying it as the first eigenvalue. Similarly $P(301/100)=-40999/10^6<0$ and $P(13/4)=41/64>0$ give a root above $3.01$, violating the last printed interval. The approximate three roots are $1.02675615$, $1.94385374$ and $3.02939011$.

There is nevertheless a corrected enclosure proving the claimed reality. Under the stated condition write $R=ϵ+\delta <(n-1)/2$. The large first row disc is disjoint from the large last column disc, since their centres are $n-1$ apart and their radii are both $R$. Every eigenvalue lies both in the row union and in the column union. It cannot belong to both large discs, so it lies in at least one of the fixed small discsThese fixed discs are pairwise disjoint. At $\delta =0$ each contains one eigenvalue. Eigenvalues depend continuously, as a multiset, on the entries; throughout the connected allowed interval no eigenvalue can move between these separated fixed sets while remaining in their union. Thus each still contains one, counted with multiplicity. Complex-conjugate pairing now proves each is real. The valid conclusion isThe source's stronger width $ϵ$ at both endpoints does not persist after adding the corner entry; it was valid for the unperturbed matrix only.

The enclosure method is Complementary row and column Gershgorin isolation.