两个正定字母的对称字方程
Symmetric Word Equations in Two Positive Definite Letters
摘要：
两个正定矩阵A和B中的广义字是A和B的非零实幂的有限乘积。正定A和B中的对称词是正定的，因此对于固定B，我们可以将对称词S（A，B）视为从正定矩阵集合到其自身的映射。给定正定的P，B和一个对称字S（A，B），A的幂为正，我们定义一个对称字方程为形式S（A，B）= P的方程.如果对任意给定的B和P，总有一个正定解A，则这样的方程是可解的.我们证明了所有对称字方程都是可解的.还讨论了这一事实的应用、求解方法、有关唯一可解性（内射性）的问题以及推广。
英文摘要：
A generalized word in two positive definite matrices A and B is a finite product of nonzero real powers of A and B. Symmetric words in positive definite A and B are positive definite, and so for fxed B, we can view a symmetric word, S(A,B), as a map from the set of positive definite matrices into itself. Given positive definite P, B, and a symmetric word, S(A,B), with positive powers of A, we defne a symmetric word equation as an equation of the form S(A,B) = P. Such an equation is solvable if there is always a positive definite solution A for any given B and P. We prove that all symmetric word equations are solvable. Applications of this fact, methods for solution, questions about unique solvability (injectivity), and generalizations are also discussed.
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