Elliptic equations and products of positive definite matrices

We present necessary and sufficient conditions under which the symmetrized product of two n x n positive definite Hermitian matrices is still a positive definite matrix (Part I, Sections 2 and 3). These results are then applied to prove the validity of the strong maximum principle, as well as of the compact support principle, for nonnegative C-1 distribution solutions of general quasilinear inequalities, possibly not elliptic at points where the gradient variable is either zero or large (Part III, Sections 9 and 10). In Part II (Sections 4-8) we consider the general problem of finding bounds for the least and greatest eigenvalues of the product of two (not necessarily definite) Hermitian matrices. In particular, we refine earlier results of Strang for this problem. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.