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

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.