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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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