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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 $$p_i,q_i\in {\mathbb R}^*$$, $$i=1,\ldots ,k$$, $$p_{k+1}\in {\mathbb R}$$. “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.

##### MSC:
 15A24 Matrix equations and identities 15B57 Hermitian, skew-Hermitian, and related matrices 15A18 Eigenvalues, singular values, and eigenvectors 15A90 Applications of matrix theory to physics (MSC2000)
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##### References:
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