Variations on a theme of Frobenius about almost commuting unitary matrices (Q1779249)

For two rotations \(A\) and \(B\) in \({\mathbb R}^n\), where \(B\) does not rotate any vector by an angle of \(\frac{\pi}{2}\) radius, \textit{F. G. Frobenius} [Sitzungsber. Akad. Wiss. Berlin, 654--665 (1911; JFM 42.0153.01)] proved that if \(A\) commutes with the commutator \( [ A,B ] = A^{-1}B^{-1}AB\) then \(A\) and \(B\) are commuting rotations. Here, the Frobenius' theorem is generalized for almost commuting unitary matrices. The proof of this generalized theorem is based on the fact that if two unitary matrices \(A\) and \(C\) almost commute, then there exists an explicit construction of a unitary change of the basis \(V\) so that \(V^{*}AV\) and \(V^{*}CV\) are simultaneously almost diagonal. The presented results with estimations of the error terms could be of interest in numerical matrix computations.