# Jordan form of the matrices of a group

Let's consider a set of $$m$$ generic square matricies $$(N;N)$$ defined on $$R$$ which forms a group. Chosen one of these $$m$$ matrices, I know that, by changing the base on my vectorial space, I can obtain a diagonal matrix or at least a matrix in a Jordan form. My question is, does exist a particular change of basis rapresented by the matrix $$T$$, which diagonalizes or puts in Jordan form all the matrices of the group? If so, how is this matrix $$T$$?

• I think at least the group should be Abelian. – xbh Nov 24 '18 at 16:57