Low rank perturbations and the spectrum of a tridiagonal sign pattern

Abstract

The n-by- n tridiagonal sign pattern T n 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 A n having the sign pattern T n are proved using new techniques on low rank perturbations. It is also shown (by using MAPLE) that for 8⩽ n⩽16, T n allows any spectrum. These results extend those previously in the literature, and strengthen the conjecture that T n 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.