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Zbl 1007.68139
Johnson, Charles R.; Hillar, Christopher J.
Eigenvalues of words in two positive definite letters. (English)
[J] SIAM J. Matrix Anal. Appl. 23, No.4, 916-928 (2002). ISSN 0895-4798; ISSN 1095-7162/e

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 2000:: *68R15 Combinatorics on words
81Q99 General mathematical topics and methods in quantum theory
20F10 Decision problems (group theory)
15A42 Inequalities involving eigenvalues and eigenvectors
15A23 Factorization of matrices
15A90 Appl. of matrix theory to physics
15A57 Other types of matrices

Keywords: positive definite matrix; word; nearly symmetric; positive eigenvalues; palindrome; trace conjecture

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