Positive eigenvalues and two-letter generalized words

Details Positive eigenvalues and two-letter generalized words Positive eigenvalues and two-letter generalized words

2005-04-28

arxiv.org/abs/math/0504573v1

Mathematics

Operator Algebras

6 Pages, Electronic Journal of Linear Algebra

Scientific paper

A generalized word in two letters $A$ and $B$ is an expression of the form $W=A^{\alpha_1}B^{\beta_1}A^{\alpha_2}B^{\beta_2}... A^{\alpha_N}B^{\beta_N}$ in which the exponents $\alpha_i$, $\beta_i$ 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.

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