Summary: A generalized word in two letters A and B is an expression of the form W = Ao""1Br'1Ao""2Br'2 * * * Ao""N Br'N in which the exponents are nonzero real numbers. When independent positive definite matrices are substituted for A and B, it is of interest whether W necessarily has positive eigenvalues. This is known to be the case when N = 1 and has been studied in case all exponents are positive by two of the authors. When the exponent signs are mixed, however, the situation is quite different (even for 2-by-2 matrices), and this is the focus of the present work.

15A18, 15A57

positive definite matrices, projections, generalized word