Uniqueness of positive definite solutions of two-letter word equations in order two
A word $W(X, B)$ is a finite product of the letters $X$ and $B$, evaluated in the written order, and it is symmetric if its sequence of letters equals its reversal. Hillar and Johnson proved that for every symmetric word $W$ in which $X$ occurs and all Hermitian positive definite $B, P \in \mathbb{C}^{n \times n}$, the equation $W(X, B) = P$ has a Hermitian positive definite solution $X$, and a real one when $B$ and $P$ are real. Armstrong and Hillar proved that for $W = XBX^2B^3X^2BX$ the real positive definite solution is unique when $B$ and $P$ are real positive definite $2 \times 2$ matrices, while for some real positive definite $3 \times 3$ matrices $B$ and $P$ this equation has more than one positive definite solution. Lawson and Lim proved uniqueness for words in which $X$ occurs at most five times.
Problem. For every symmetric word $W(X, B)$ in which $X$ occurs at least once and all Hermitian positive definite $B, P \in \mathbb{C}^{2 \times 2}$, is there exactly one Hermitian positive definite $X \in \mathbb{C}^{2 \times 2}$ with $W(X, B) = P$?
Source: Hillar and Johnson (2004), Conjecture 2.3, in every order and with real exponents; for order two, Armstrong and Hillar (2007), Conjecture 11.5, for words in several fixed letters. Stated for ordinary words in two letters by OpenProblemsInNLA.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- RA
- Keywords
- nonlinear matrix equations, OpenProblemsInNLA, positive definite matrices, word equations
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@article{armstrong2007,
author = {Armstrong, Scott N. and Hillar, Christopher J.},
title = {Solvability of symmetric word equations in positive definite letters},
journal = {J. London Math. Soc. (2)},
volume = {76},
number = {3},
year = {2007},
pages = {777--796},
doi = {10.1112/jlms/jdm070},
eprint = {math/0507306},
archivePrefix = {arXiv},
}
@article{hillar2004,
author = {Hillar, Christopher J. and Johnson, Charles R.},
title = {Symmetric word equations in two positive definite letters},
journal = {Proc. Amer. Math. Soc.},
volume = {132},
number = {4},
year = {2004},
pages = {945--953},
doi = {10.1090/S0002-9939-03-07163-6},
eprint = {math/0209399},
archivePrefix = {arXiv},
}
@article{lawson2006,
author = {Lawson, Jimmie and Lim, Yongdo},
title = {Solving symmetric matrix word equations via symmetric space machinery},
journal = {Linear Algebra Appl.},
volume = {414},
number = {2-3},
year = {2006},
pages = {560--569},
doi = {10.1016/j.laa.2005.10.035},
}
@misc{openproblemsinnla-mf16,
author = {Townsend, Alex},
title = {{MF-16}: Uniqueness of positive definite solutions of two-letter word equations in order two},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-functions-and-stability/MF-16/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
The symmetric word equation $XBX^{12}BX = P$, with $B = \begin{pmatrix} 1 & 4 \\ 4 & 17 \end{pmatrix}$ and an explicit integer matrix $P$, has at least three distinct real symmetric positive definite $2 \times 2$ solutions. At the solution $X_0 = \operatorname{diag}(3, 1)$ the Jacobian determinant of the word map in symmetric-matrix coordinates is exactly $-11785057051824$, so Armstrong …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Needs verification → Solution claimed | clemens-admin | Listed with the record of its solution, unreviewed. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00233, "Uniqueness of positive definite solutions of two-letter word equations in order two", clopen.solutions.