On Nonnegative matrices generating a finite multiplicative monoid

Abstract

Square nonnegative matrices with the property that the multiplicative monoid M(A) generated by the matrix A is finite are characterized in several ways. At first, the least general upper bound for the cardinality of M(A) is derived. Then it is shown that any square nonnegative matrix is cogredient to a lower triangular block form with the diagonal consisting of three blocks L, A ₀, and M where L and M are strictly lower triangular, A ₀ has no zero rows or columns, and M(A) is finite if and only if. M(A ₀) is so. Several criteria for, M(A ₀) to be finite are presented. One of the normal forms of A applies very well to the characterization of the nonnegative solutions of each of the matrix equations X ^k = 0, X ^k = 1, X ^k = X, and X ^k = X ^T where k > 1 is an integer. It also leads to a polynomial time algorithm for deciding whether or not M(A) is finite, if the entries of A are nonnegative rationals.

Keywords:

• Nonnegative matrix
• monoid
• Perron-Frobenius theorem
• block form
• matrix equation