Symmetric Word Equations in Two Positive Definite Letters

Details Symmetric Word Equations in Two Positive Definite Letters Symmetric Word Equations in Two Positive Definite Letters

2002-09-29

arxiv.org/abs/math/0209399v1

Mathematics

Rings and Algebras

9 pages

Scientific paper

A generalized word in two positive definite matrices A and B is a finite product of nonzero real powers of A and B. Symmetric words in positive definite A and B are positive definite, and so for fxed B, we can view a symmetric word, S(A,B), as a map from the set of positive definite matrices into itself. Given positive definite P, B, and a symmetric word, S(A,B), with positive powers of A, we defne a symmetric word equation as an equation of the form S(A,B) = P. Such an equation is solvable if there is always a positive definite solution A for any given B and P. We prove that all symmetric word equations are solvable. Applications of this fact, methods for solution, questions about unique solvability (injectivity), and generalizations are also discussed.

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