Symmetric Word Equations in Two Positive Definite Letters
نویسندگان
چکیده
For every symmetric (“palindromic”) word S(A,B) in two positive definite letters and for each fixed n-by-n positive definite B and P , it is shown that the symmetric word equation S(A,B) = P has an n-by-n positive definite solution A. Moreover, if B and P are real, there is a real solution A. The notion of symmetric word is generalized to allow non-integer exponents, with certain limitations. In some cases, the solution A is unique, but, in general, uniqueness is an open question. Applications and methods for finding solutions are also discussed.
منابع مشابه
ar X iv : m at h / 02 09 39 9 v 1 [ m at h . R A ] 2 9 Se p 20 02 SYMMETRIC WORD EQUATIONS IN TWO POSITIVE DEFINITE LETTERS
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 fixed 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, ...
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