The Elementary Divisors, Associated with 0, of a Singular M-matrix

Abstract

1. Many investigations have been concerned with a squaro matrix P with non-negative coefficients (elements). It is remarkable that many interesting properties of P are determined by the set Σ of index pairs of positive ( i.e. non-zero) coefficients of P , the actual values of these coefficients being irrelevant. Thus, for example, the number of characteristic roots equal in absolute value to the largest non-negative characteristic root p depends on Σ alone, if P is irreducible. If P is reducible, then Σ determines the standard forms of P (cf. § 3). The multiplicity of p depends on Σ, and on the set S of indices of those submatrices in the diagonal in a standard form of P which have p as a characteristic root. It has apparently not been considered before whether Σ and S also determine the elementary divisors associated with p . We shall show that, in general, the elementary divisors do not depend on these sets alone, but that necessary and sufficient conditions may be found in terms of Σ and S ( a ) for the elementary divisors associated with p to be simple, and ( b ) that there is only one elementary divisor associated with p .