Solving Symmetric Word Equations for Positive Definite Letters:
                                      Theory and Practice
                     by Charles R. Johnson


                           ABSTRACT
     By a symmetric word in two letters A and B we mean the juxtaposition of a 
finite sequence of A's and B's that reads the same right-to-left as 
left-to-right. We interpret A and B as independent positive definite (PD) 
matrices and juxtaposition as matrix multiplication. If S(A,B) is a symmetric 
word in two PD letters A and B, it is easily seen that P = S(A,B) is also PD. 
By a symmetric word equation, we mean one of the form
                         P = S(A,B)
in which P and B are given PD matrices. We discuss the solvability of 
symmetric word equations for a PD matrix A and describe an algorithm for 
solving for A. Such word equations arise in the study of a long standing 
problem from quantum physics, and the current work is joint with Chris Hillar 
(UC, Berkeley).