Compartmental matrices with single root and nonnegative nilpotent matrices

Abstract

It is shown that, as a consequence of the Frobenius Theorem on nonnegative matrices, an n-square compartmental matrix with a single n-fold root is permutationally similar to a triangular matrix. As a result, certain properties of precedence matrices and some graph theoretic results can be viewed as essentially resulting from the classical theorem on nonnegative irreducible matrices.