Almost commuting matrices in the rank metric
If two invertible matrices almost commute in the normalised rank metric, are they close to a pair of invertible matrices that commute exactly?
Cite
@misc{cairn-almost-commuting-matrices-rank-metric,
title = {Almost commuting matrices in the rank metric},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/almost-commuting-matrices-rank-metric}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
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The problem
The question
For n×n matrices over a field F, the normalised rank distance is d(A, B) = rank(A − B)/n. Suppose invertible matrices A_n, B_n satisfy d(A_n B_n, B_n A_n) → 0. Must there be commuting invertible matrices A'_n, B'_n with d(A_n, A'_n) → 0 and d(B_n, B'_n) → 0? (In the language of stability: is Z^2 stable with respect to the rank metric on GL_n(F)?)
What is known
For permutations with the Hamming metric the answer is yes (Arzhantseva–Păunescu). Over the complex numbers there are positive answers for unitary or self-adjoint tuples (Elek–Grabowski), but the general invertible case is explicitly left open, as is rank-stability of Z^k without spectral restrictions.
What counts as progress
- Proofs for further classes (normal matrices, specific fields, finite fields).
- Candidate counterexample families with computed lower bounds on the distance to commuting pairs.
- A complete answer.
Source. Posed by Goulnara Arzhantseva in the open problem session of the Oberwolfach workshop Mini-Workshop: Growth and Expansion in Groups (2024), recorded in Oberwolfach Reports 17/2024, p. 1035 (EMS Press, DOI 10.4171/OWR/2024/17), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.