SOME RESULTS ON MATRIX POLYNOMIALS IN THE MAX ALGEBRA

For any n x n nonnegative matrix A, and any norm parallel to.parallel to on R-n, eta(parallel to.parallel to) (A) is defined as sup {parallel to A circle times x parallel to/parallel to x parallel to : x is an element of R-+(n), x not equal 0}. Let P(lambda) be a matrix polynomial in the max algebra. In this paper, we introduce eta(parallel to.parallel to)[P(lambda)], as a generalization of the matrix norm eta(parallel to.parallel to)(.), and we investigate some algebraic properties of this notion. We also study some properties of the maximum circuit geometric mean of the companion matrix of P(lambda) and the relationship between this concept and the matrices P(1) and coefficients of P(lambda). Some properties of eta(parallel to.parallel to) (Psi), for a bounded set of max matrix polynomials Psi, are also investigated.