On the convergence of the ordered roots of a sequence of determinantal equations

Abstract

Let λ 1( n)⩾λ 2( n)⩾⋯⩾λ p ( n) be the ordered roots of | A( n)−λ B( n)| = 0, where A( n) is a p × p real symmetric matrix and B( n) is a p p real symmetric positive definite matrix. Assume that A( n) → A and B( n) → B as n → ∞, where the rank of B may be less than p. Using a result on the continuity of the zeros of a polynomial, the limiting behavior of p sequences {λ j ( n)} ∞ n = 1 , j = 1, 2,…, p, is investigated.