Latent Roots of Tri-Diagonal Matrices

Abstract

A considerable amount is known about the latent roots of matrices of the form in the case when each cross-product of non-diagonal elements, a i c i-1 , is positive. One forms the sequence of polynomials f r (λ) = |L r −λI| for r = 1, 2, … n , and observes that then it is easy to deduce that (i) the zeros of f n (λ) and f n_1 (λ) interlace—that is, between two consecutive zeros of either polynomial lies precisely one zero of the other (ii) at the zeros of f n (λ) the values of f n-x (λ) are alternately positive and negative, (iii) all the zeros of f n (λ) — i.e. all the latent roots of L n —are real and different.