Almost commuting matrices with respect to the rank metric (Q2076065)

This article discusses a variant of the famous Halmos problem: to perturb two almost commuting matrices such that their approximate versions are commuting. In general, not all such matrix perturbations exist. The answer to the Halmos problem is positive when the two given matrices are self-adjoint and with norm one. However, the answer is negative when the two matrices are unitary. Using the different distance metrics, such as the Hamming distance or the normalized Hilbert-Schmidt norm for defining the closeness of two matrices, the answer to the above problem is also positive, as proved by \textit{P. Rosenthal} [Amer. Math. Monthly 76, No. 8, 925--926 (1969)], \textit{G. Arzhantseva} and \textit{L. Păunescu} [J. Funct. Anal. 269, No. 3, 745--757 (2015; Zbl 1368.20025)]. Instead of using the usually norm-induced metric, the present authors propose a rank metric to measure the closeness of a matrix (unitary or self-adjoint) and its perturbed version. Like the Hamming distance, the rank distances have been used widely in coding theory. This paper might find applications in error-correcting codes, e.g., rank codes, which attracted the attention partly thanks to the work by \textit{È. M. Gabidulin} [Probl. Inf. Transm. 21, 1--12 (1985; Zbl 0585.94013); translation from Probl. Peredachi Inf. 21, No. 1, 3--16 (1985)]. Also interesting is the conclusion that Abel's group is not stable with the rank metric. The result might serve as a reminder that more attention should be taken when constructing well-behaved rank codes.