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 | References | Similar Articles | Additional Information

Abstract: For every symmetric (``palindromic") word in two positive definite letters and for each fixed -by- positive definite and , it is shown that the symmetric word equation has an -by- positive definite solution . Moreover, if and are real, there is a real solution . The notion of symmetric word is generalized to allow non-integer exponents, with certain limitations. In some cases, the solution is unique, but, in general, uniqueness is an open question. Applications and methods for finding solutions are also discussed.

**1.**T. Ando,*On the arithmetic-geometric-harmonic-mean inequalities for positive definite matrices*, Linear Algebra Appl.**52/53**(1983), 31–37. MR**709342**, 10.1016/0024-3795(83)80005-6**2.**Roger A. Horn and Charles R. Johnson,*Matrix analysis*, Cambridge University Press, Cambridge, 1985. MR**832183****3.**Roger A. Horn and Charles R. Johnson,*Topics in matrix analysis*, Cambridge University Press, Cambridge, 1991. MR**1091716****4.**Charles R. Johnson and Christopher J. Hillar,*Eigenvalues of words in two positive definite letters*, SIAM J. Matrix Anal. Appl.**23**(2002), no. 4, 916–928 (electronic). MR**1920925**, 10.1137/S0895479801387073**5.**Eberhard Zeidler,*Applied functional analysis*, Applied Mathematical Sciences, vol. 108, Springer-Verlag, New York, 1995. Applications to mathematical physics. MR**1347691**

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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:
https://doi.org/10.1090/S0002-9939-03-07163-6

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