Eigenvalues of Words in Two Positive Definite Letters
Hillar, Christopher J
Johnson, Charles R
Mathematics - Operator Algebras | Mathematics - Rings and Algebras | 15A24, 15A57 | 15A18, 15A90
arxiv: Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) | Computer Science::Formal Languages and Automata Theory
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.