Abstract. In this paper we give a matrix version of Handelman's Positivstellensatz [4], representing polynomial matrices which are positive definite on convex, compact polyhedra.Moreover, we propose also a procedure to find such a representation. As a corollary of Handelman's theorem, we give a special case of Schmüdgen's Positivstellensatz for polynomial matrices positive definite on convex, compact polyhedra.

A number λ ∈ C is called an eigenvalue of the matrix polynomial P(z) if there exists a nonzero vector x ∈ C n such that P(λ)x = 0. Note that each finite eigenvalue of P(z) is a zero of the characteristic polynomial det(P(z)). In this paper we establish some (upper and lower) bounds for eigenvalues of matrix polynomials based on the norm of their coefficient matrices and compare these bounds to those given by N. J. Higham and F. Tisseur [8], J. Maroulas and P. Psarrakos [12].

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