Solvability of Symmetric Word Equations in Positive Definite Letters

Details Solvability of Symmetric Word Equations in Positive Definite Letters Solvability of Symmetric Word Equations in Positive Definite Letters

2005-07-15

arxiv.org/abs/math/0507306v4

Mathematics

Operator Algebras

21 pages

Scientific paper

Let $S(X,B)$ be a symmetric (``palindromic'') word in two letters $X$ and $B$. A theorem due to Hillar and Johnson states that for each pair of positive definite matrices $B$ and $P$, there is a positive definite solution $X$ to the word equation $S(X,B)=P$. They also conjectured that these solutions are finite and unique. In this paper, we resolve a modified version of this conjecture by showing that the Brouwer degree of such an equation is equal to 1 (in the case of real matrices). It follows that, generically, the number of solutions is odd (and thus finite) in the real case. Our approach allows us to address the more subtle question of uniqueness by exhibiting equations with multiple real solutions, as well as providing a second proof of the result of Hillar and Johnson in the real case.

Affiliated with

Also associated with

No associations

Rating

Solvability of Symmetric Word Equations in Positive Definite Letters does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Solvability of Symmetric Word Equations in Positive Definite Letters, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Solvability of Symmetric Word Equations in Positive Definite Letters will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-411660