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.