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Solving symmetric matrix word equations via symmetric space machinery. (English) Zbl 1105.15013

For a non-empty alphabet $𝒜=\left\{{A}_{1},{A}_{2},\cdots \right\}$ the authors consider a semigroup with identity of generalized words of the form $W:=W\left({A}_{1},\cdots ,{A}_{k}\right)={A}_{1}^{{p}_{1}}\cdots {A}_{k}^{{p}_{k}}$, where ${p}_{j}$ are real numbers. The word $W$ is symmetric if it is equal to ${A}_{k}^{{p}_{k}}\cdots {A}_{1}^{{p}_{1}}$. A symmetric word equation for $𝐀=𝒜\cup \left\{X,B\right\}$ is an equation of the form $W\left(X,{A}_{1},\cdots ,{A}_{k}\right)=B$, where $W\left(X,{A}_{1},\cdots ,{A}_{k}\right)$ is a symmetric word in $X,{A}_{1},\cdots ,{A}_{k}$, all exponents of $X$ are positive, and all exponents of ${A}_{j}$ are non-negative. Symmetric word equations arised naturally in matrix theory as equations over the cone of positive definite matrices. A symmetric word equation $W\left(X,A\right)=B$ is called (uniquely) solvable if there exists a (unique) positive definite solution $X$ of $W\left(X,A\right)=B$ for any pair of $n×n$ positive definite matrices $A$ and $B$. It was proved [see C. J. Hillar and C. R. Johnson, Proc. Am. Math. Soc. 132, 945–953 (2004; Zbl 1038.15005)], that every positive definite word equation is solvable.

The authors investigate the uniqueness conjecture and the continuity of solutions as a function on the variables $A$ and $B$ over positive definite matrices. The main goal of the paper is threefold.

1. The authors demonstrate how the geometric mean of two matrices and its generalizations to weighted means can be used to give explicit solutions to certain classes of equations.

2. It is shown how the geometry of the positive definite matrices equipped with a symmetric structure and a convex Riemannian metric allows to deduce solution for other classes of symmetric equations via the application of the Banach fixed point theorem.

3. It is shown that equations through degree 5 are uniquely solvable and the solution is continuous in $A$ and $B$. It is noted that the degree 5 is the best possible since it was already shown that there are degree 6 equations possessing with multiple solutions.

##### MSC:
 15A24 Matrix equations and identities