## Symmetric word equations in two positive definite letters

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- by Christopher J. Hillar and Charles R. Johnson PDF
- Proc. Amer. Math. Soc.
**132**(2004), 945-953 Request permission

## 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.## References

- T. Ando,
*On the arithmetic-geometric-harmonic-mean inequalities for positive definite matrices*, Linear Algebra Appl.**52/53**(1983), 31–37. MR**709342**, DOI 10.1016/0024-3795(83)80005-6 - Roger A. Horn and Charles R. Johnson,
*Matrix analysis*, Cambridge University Press, Cambridge, 1985. MR**832183**, DOI 10.1017/CBO9780511810817 - Roger A. Horn and Charles R. Johnson,
*Topics in matrix analysis*, Cambridge University Press, Cambridge, 1991. MR**1091716**, DOI 10.1017/CBO9780511840371 - 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. MR**1920925**, DOI 10.1137/S0895479801387073 - Eberhard Zeidler,
*Applied functional analysis*, Applied Mathematical Sciences, vol. 108, Springer-Verlag, New York, 1995. Applications to mathematical physics. MR**1347691**

## 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
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**132**(2004), 945-953 - MSC (2000): Primary 15A24, 15A57; Secondary 15A18, 15A90
- DOI: https://doi.org/10.1090/S0002-9939-03-07163-6
- MathSciNet review: 2045408