Remarks on Strongly Hyperbolic Matrices

The paper is devoted to studying uniformly strongly hyperbolic matrices P(z,ξ), where z ∈ <b>R</b>^d and ξ ∈ <b>R</b>ⁿ. It is proved that if the characteristic roots of P(z,ξ) are outside a strip of the form |Im$\tau$|< δ, then there is a Hermitian smooth matrix function Q(z,ξ) with eigenvalues separated from zero uniformly with respect to (z,ξ) such that i(P*Q-QP) \ge Q. To establish this assertion, we refine well-known results on the transformation of homogeneous and nonhomogeneous hyperbolic matrices to the diagonal and block-diagonal forms, respectively. The results obtained in this article will be used in forthcoming papers to investigate large-time behavior of solutions to first-order strongly hyperbolic systems.