CHARACTERISTIC ROOTS OF M-MATRICES

A square matrix B is called a nonnegative matrix (written B ?0) if each element of B is a nonnegative number. It is well known [II that every nonnegative matrix B has a nonnegative characteristic root p(B) (the Perron root of B) such that each characteristic root X of B satisfies |X| ?p(B). A square matrix A is called an M-matrix if it has the form k -I-B, where B is a nonnegative matrix, k > p(B), and I denotes the identity matrix. In case A is a real, square matrix with nonpositive off-diagonal elements, each of the following is a necessary and sufficient condition for A to be an M-matrix [4, p. 387]. (1) Each principal minor of A is positive. (2) A is nonsingular, and A-1 ?0. (3) Each real characteristic root of A is positive. (4) There is a row vector x with positive entries (x>O) such that xA > 0. If A is an M-matrix, then A has a positive characteristic root q(A) which is minimal, in the sense that for each characteristic root : of A, q(A) ? 1 [4, p. 389]. Bounds for q(A) can be readily obtained using known bounds for the Perron root of a nonnegative matrix. In this paper, as in [2], we reverse this procedure, studying the characteristic roots of M-matrices in order to find new bounds for Perron roots. Ky Fan proved the following lemma [3]: