Almost commuting matrices with respect to the rank metric
G\'abor Elek, {\L}ukasz Grabowski
TL;DR
This paper demonstrates that matrices which are nearly commuting and are either unitary or self-adjoint can be approximated by exactly commuting matrices close in the rank metric, advancing understanding of matrix approximations.
Contribution
It introduces a new approximation result for matrices nearly commuting in the rank metric, specifically for matrices that are unitary or self-adjoint.
Findings
Nearly commuting matrices can be approximated by commuting matrices in the rank metric.
The approximation applies to matrices that are either unitary or self-adjoint.
The result extends the theory of matrix approximations in non-commutative settings.
Abstract
We show that if A_1, A_2, ... , A_n are square matrices, each of them is either unitary or self-adjoint, and they almost commute with respect to the rank metric, then one can find commuting matrices B_1, B_2, ... , B_n that are close to the matrices A_i in the rank metric.
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