Positive eigenvalues and two-letter generalized words (Q2778503)

A so-called generalized word of class \(N\) in two letters \(A\) and \(B\) is defined as an expression \(W(A,B)=A^{\alpha_1}B^{\beta_1}A^{\alpha_2} B^{\beta_2}\dots A^{\alpha_N}B^{\beta_N}\), where the exponents \(\alpha_1, \beta_1,\dots,\alpha_N,\beta_N\) are nonzero real numbers. Here, the eigenvalues of words \(W(A,B)\) are studied when \(A\) and \(B\) are replaced by positive definite \(n \times n\) matrices. It is of special interest to know when words \(W(A,B)\) have positive eigenvalues. NEWLINENEWLINENEWLINEIn the previously studied case when \(N=1\) and all exponents were positive the eigenvalues were positive. In this work this case is extended on the situation when the exponents have mixed signs. In several theorems conditions on \(A\), \(B\) (e. g. multiplicity of eigenvalues) and on a sign pattern of exponent sequences for the cases \(n=2,3\) and arbitrary \(n\) for \(W(A,B)\) to have positive eigenvalues are found.