Positive eigenvalues and two-letter generalized words

2005/04/28 by Christopher Hillar, Christopher J. Hillar, Hillar, Christopher +4
Computer Science · Mathematics · Physics and Astronomy · #15A18 #15A57 #FOS: Mathematics #Matrix Theory and Algorithms #Operator Algebras (math.OA) #Quantum chaos and dynamical systems #Rings and Algebras (math.RA) #Spectral Theory in Mathematical Physics #math.OA #math.RA #msc:15A18 #msc:15A57

paper · pdf · doi:10.48550/arxiv.math/0504573

6 Pages, Electronic Journal of Linear Algebra

arxiv created 2005/04/28 · openalex publication_date 2005/04/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A generalized word in two letters A and B is an expression of the form W=A^α₁B^β₁A^α₂B^β₂... A^α_NB^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.

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