Low rank perturbations and the spectrum of a tridiagonal sign pattern

The n-by-n tridiagonal sign pattern Tn has every superdiagonal entry positive, every subdiagonal entry negative, the (1,1) entry negative, the (n,n) entry positive and every other diagonal entry zero. Inertia and spectral results for matrices An having the sign pattern Tn are proved using new techniques on low rank perturbations. It is also shown (by using MAPLE) that for 8 n 16, Tn allows any spectrum. These results extend those previously in the literature, and strengthen the conjecture that Tn allows any spectrum for all values of n. In addition, bounds on the algebraic multiplicity of an eigenvalue of a low rank perturbation of a general matrix are obtained.