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Eigenvalues of words in two positive definite letters. (English) Zbl 1007.68139
Summary: The question of whether all words in two real positive definite letters have only positive eigenvalues is addressed and settled (negatively). This question was raised some time ago in connection with a long-standing problem in theoretical physics. A large class of words that do guarantee positive eigenvalues is identified, and considerable evidence is given for the conjecture that no other words do. In the process, a fundamental question about solvability of symmetric word equations is encountered.
MSC:
68R15Combinatorics on words
81Q99General mathematical topics and methods in quantum theory
20F10Decision problems (group theory); connections with logic and automata
15A42Inequalities involving eigenvalues and eigenvectors
15A23Factorization of matrices
15A90Appl. of matrix theory to physics (MSC2000)
15A57Other types of matrices (MSC2000)