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

Gershgorin's theorem states that every complex eigenvalue of a square matrix lies in at least one row discLet $Av=\lambda v$ with $v\ne 0$, and choose an index $i$ where $|v_i|$ is maximal. Its row equation givesDividing by the positive $|v_i|$ proves $\lambda \in D_i$. This proof applies to complex eigenvectors even when the matrix is real.

Gershgorin's second theorem states that a union of $k$ row discs separated from the union of all remaining discs contains exactly $k$ eigenvalues, counted with algebraic multiplicity. In particular a disc disjoint from every other disc contains exactly one eigenvalue. The same assertion holds for column discs by applying it to the transpose.

Assume $0<ϵ<1/4$, as implied by its stipulated smallness, and use the nontrivial matrix range $n\ge 3$. The row radii are $ϵ$ at one, $r=1/4+ϵ$ at each interior index, and $1/4$ at $n$. Since $2r<1$, all discs are disjoint. The second theorem therefore gives one eigenvalue in each disc. Each such eigenvalue is real: a nonreal eigenvalue would have its distinct complex conjugate in the same disc, contradicting its count of one. ThusThe eigenvalues are simple as well. An alternative reality argument is that a positive diagonal similarity makes this tridiagonal matrix symmetric, since its opposite off-diagonal entries have positive product.

The transpose has first radius $1/4$, interior radii $1/4+ϵ$, and last radius $ϵ$. These discs are again disjoint and each has one eigenvalue. Intersecting with the previous enclosures therefore improves the last bound toThe identification with the same last eigenvalue is justified by separation from all the other integer-centred discs.

Put $R=ϵ+\delta$. Only the first row disc changes: it now has radius $R$. The other discs keep radii $r_i=1/4+ϵ$ for $2\le i\le n-1$ and $r_n=1/4$. They remain mutually disjoint. As $R$ grows, the first disc joins an initial segment of these discs. Define $m$ as the largest index joined to it, or one if no other disc is joined. The joining condition for index $i$ isFor an interior index this starts at $\delta =i-5/4-2ϵ$, and for the final index at $\delta =n-5/4-ϵ$. Until its joining threshold, disc $i$ contains a unique real eigenvalue and retains the bound $|{\lambda }_i-i|\le r_i$.

After the first $m$ discs have joined, the second theorem places exactly $m$ eigenvalues in their union and exactly one in each remaining disc. The cluster is bounded, for instance, byand by $|z-1|\le R$ when $m=1$. The union of the actual discs is a sharper enclosure. Once discs overlap, one cannot assign one eigenvalue to each individual disc merely by its centre; the theorem counts the whole separated cluster.

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.