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).