Simple algorithm for correction of a finite matrix group known approximately

Abstract

In this paper we have considered a finite unitary matrix group with exact elements being unknown and only approximate elements available. Such a group becomes inconsistent with its own multiplication table. We found simple correction formula for such group. When applied iteratively this formula gives fast convergence of the group elements and allows to perform the group reconstruction. Next we considered small unitary rotation of entire group which makes the group consistent with a set of additional conditions. For example, we might demand that group elements act on certain set of vectors in a predefined manner. Again, iterative procedure based on this correction shows fast convergence. All algorithms developed in this paper were implemented in a Python library which is available as open source software.