ELA, Volume 9, pp. 21-26, February 2002, abstract.

Positive eigenvalues and two-letter generalized words

C. Hillar, C.R. Johnson, and I.M. Spitkovsky

A generalized word in two letters A and B is an 
expression of the form 
  W = A^{a1} B^{b1} A^{a2} B^{b2} ... A^{aN} B^{bN} 
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.