Allowable spectral perturbations for ZME-matrices (Q1112138)

Let A be a real square matrix. Then A is called a ZM-matrix if the off- diagonal entries of each positive power of A are nonpositive. This class of matrices has been studied recently by \textit{D. Hershkowitz} and the first author [Isr. J. Math. 55, 327-344 (1986; Zbl 0625.15017)] and \textit{S. Friedland}, \textit{D. Hershkowitz} and the first author [Trans. Am. Math. Soc. 300, 343-366 (1987; Zbl 0619.15018)]. In the present paper the authors introduce the class of ZME-matrices. A ZM-matrix is called a ZME-matrix if all its odd powers are irreducible, and either all its even powers are irreducible or all its even powers are completely reducible but not irreducible. This long paper is an investigation of ZME-matrices, particularly their spectral properties. The results are too technical to state, but a principal theme is the determination of conditions under which perturbations of ZME-matrices remain ZME-matrices. The last section of the paper deals with simple conditions under which the product of two commuting ZME-matrices is again a ZME-matrix.