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Zbl 1038.15005
Hillar, Christopher J.; Johnson, Charles R.
Symmetric word equations in two positive definite letters.
(English)
[J] Proc. Am. Math. Soc. 132, No. 4, 945-953 (2004). ISSN 0002-9939; ISSN 1088-6826/e

Consider the word equation $S(A,B)=P$ where $A,B,P$ are positive definite complex Hermitian $n\times n$-matrices, $A$ is the unknown matrix, and $S(A,B)$ is a symmetric (palindromic'') generalized word of the form $W=A^{p_1}B^{q_1}\ldots A^{p_k}B^{q_k}A^{p_{k+1}}$; here $p_i,q_i\in {\Bbb R}^*$, $i=1,\ldots ,k$, $p_{k+1}\in {\Bbb R}$. Symmetric'' means that $W=A^{p_{k+1}}B^{q_k}A^{p_k}\ldots B^{q_1}A^{p_1}$. \par The authors show that every symmetric word equation is solvable and they conjecture uniqueness of the solution. In some cases the solution is unique (example: the unique solution of the equation $ABA=P$ is $A=B^{-1/2}(B^{1/2}PB^{1/2})^{1/2}B^{-1/2}$); in general, uniqueness is an open question. If $B$ and $P$ are real, then one can find a real solution $A$ as well. The notion of symmetric word is generalized to allow non-integer exponents, with certain limitations. Applications and methods for finding solutions are also discussed.
MSC 2000:
*15A24 Matrix equations
15A57 Other types of matrices
15A18 Eigenvalues of matrices, etc.
15A90 Appl. of matrix theory to physics

Keywords: positive definite matrix; generalized word; symmetric word equation

Cited in: Zbl 1134.15009 Zbl 1105.15013

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