Generalized eigenvectors and sets of nonnegative matrices

In this paper we present extensions of the Perron-Frobenius theory for square irreducible nonnegative matrices. After showing a generalization to reducible matrices, we extend the theory to sets of nonnegative matrices, which play an important role in several dynamic programminq recursions (e.q. Markov decision processes) and in mathematical economics (e.q. Leontief substitution systems). We consider a finite set M of (in general reducible) matrices, which is generated by all possible interchanges of corresponding rows, selected from a fixed finite set of square nonnegative matrices. A simultaneous block-triangular decomposition of the set of matrices will be presented and characterized in terms of the maximal spectral radius and the maximal index, associated with this maximal spectral radius, using the concept of generalized eigenvectors. As a by-product of our analysis we obtain a generalization of Howard's policy iteration method. This paper extends earlier results of Sladky [7] and Zijm [8].