Low rank perturbations and the spectrum of a tridiagonal sign pattern

The n-by-n tridiagonal sign pattern T-n has every superdiagonal entry positive, every sub-diagonal 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 less than or equal to n less than or equal to 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. (C) 2003 Elsevier Inc. All rights reserved.