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

Write $q=1/4+ϵ$ and take $0<ϵ<1/4$, as intended by the small-parameter assumption. The row discs have centers $1,\dots ,n$ and radii $ϵ,q,\dots ,q,1/4$. They are pairwise disjoint because the sum of neighboring radii is below one. The second Gershgorin circle theorem places one eigenvalue in each disc. Since $A$ is real, nonreal eigenvalues occur in conjugate pairs; a disc centered on the real axis would have to contain both members, contradicting its count of one. Thus all eigenvalues are real andAlternatively, a positive diagonal change of basis symmetrizes the tridiagonal matrix, with transformed off-diagonal entries $\sqrt{ϵ}/2$, also proving reality.