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Symmetric word equations in two positive definite letters. (English) Zbl 1038.15005
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 $W=A^{p_{k+1}}B^{q_k}A^{p_k}\ldots B^{q_1}A^{p_1}$. 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.

15A24Matrix equations and identities
15B57Hermitian, skew-Hermitian, and related matrices
15A18Eigenvalues, singular values, and eigenvectors
15A90Applications of matrix theory to physics (MSC2000)
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