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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Symmetric word equations in two positive definite letters


Authors: Christopher J. Hillar and Charles R. Johnson
Journal: Proc. Amer. Math. Soc. 132 (2004), 945-953
MSC (2000): Primary 15A24, 15A57; Secondary 15A18, 15A90
Published electronically: September 22, 2003
MathSciNet review: 2045408
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Abstract: 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.


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Additional Information

Christopher J. Hillar
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: chillar@math.berkeley.edu

Charles R. Johnson
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795
Email: crjohnso@math.wm.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07163-6
PII: S 0002-9939(03)07163-6
Keywords: Positive definite matrix, generalized word, symmetric word equation
Received by editor(s): June 21, 2002
Received by editor(s) in revised form: November 20, 2002
Published electronically: September 22, 2003
Additional Notes: This research was conducted, in part, during the summer of 1999 at the College of William and Mary’s Research Experiences for Undergraduates program and was supported by NSF REU grant DMS-96-19577
The work of the first author is supported under a National Science Foundation Graduate Research Fellowship
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society