Asymptotic diagonalization of matrix systems

In recent years many results have been obtained on the asymptotic behavior of solutions of the matrix difference equation Mnxn=xn+1 where {Mn}n=0∞ is a sequence of k×kk×k-matrices with real or complex entries that are close to diagonal matrices. In this paper we study the question of how to transform a matrix sequence {Mn}n=0∞ where the entries behave sufficiently regularly, into a sequence of almost-diagonal matrices, so that the results for almost-diagonal matrices can be applied to the difference equation with the transformed sequence. In particular, we will try to find explicit matrices BnBn such that the matrices Mn′=Bn+1−1MnBn are close to diagonal matrices and a Levinson-type theorem can be applied to transform the sequence {Mn′}n=0∞ into a sequence of diagonal matrices. In the case that the MnMn are real 2×2-matrices, a fairly general answer is obtained and it is shown how to proceed for a given sequence {Mn}n=0∞. Furthermore, we prove a couple of results that are useful for the case of general order kk.