Positive eigenvalues and two-letter generalized words

Abstract

A generalized word in two letters $A$ and $B$ is an expression of the form $W=A^{α_1}B^{β_1}A^{α_2}B^{β_2}... A^{α_N}B^{β_N}$ in which the exponents $α_i$, $β_i$ 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.