On the torsion in the general linear group and the diagonalization algorithm

This work describes periodic matrices in the general linear group over the real numbers field and over the maximal Abelian extension $\mathbb{Q}_{\mathrm{ab}}$ of the rational numbers field. It is shown that for the case of real numbers the general question is reduced to the $2\times2$ matrices. A simple periodicity criterion is provided for them. We demonstrate a geometric interpretation of the results. The main result is an algorithm that tests periodicity of a matrix and, if the matrix is periodic, finds its Jordan form.