Generalized eigenvectors and sets of nonnegative matrices

Abstract

We present extensions of the Perron-Frobenius theory for square irreducible nonnegative matrices. After discussing structural properties of reducible nonnegative matrices we extend the theory to sets of nonnegative matrices, which play an important role in several dynamic programming recursions (e.g. Markov decision processes) and in mathematical economics (e.g. Leontief substitution systems). A set K of (in general, reducible) matrices is considered, 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 K is presented and characterized in terms of the maximal spectral radius, the maximal index, and generalized eigenvectors. As a by-product of our analysis we obtain a generalization of Howard's policy iteration method.