Positive eigenvalues and two-letter generalized words

الأصل · EN

A generalized word in two letters A and B is an expression of the form W=Aα¹Bβ¹Aα²Bβ²... AαⁿBβⁿ in which the exponents αᵢ, βᵢ are nonzero real numbers. When independent positive definite matrices are substituted for A and B, we are interested in 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.