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

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.