Latent Roots of Tri-Diagonal Matrices

Abstract

A considerable amount is known about the latent roots of matrices of the formin the case when each cross-product of non-diagonal elements, aici-1, is positive. One forms the sequence of polynomials fr(λ) = |Lr−λI| for r = 1, 2, … n, and observes thatthen it is easy to deduce that (i) the zeros of fn(λ) and fn_1(λ) interlace—that is, between two consecutive zeros of either polynomial lies precisely one zero of the other (ii) at the zeros of fn(λ) the values of fn-x(λ) are alternately positive and negative, (iii) all the zeros of fn(λ)— i.e. all the latent roots of Ln—are real and different.