Resumen de Asymptotic diagonalization of matrix systems

R. J. Kooman

• 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 �~ 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 Bn such that the matrices M��n = B.1 n+1Mn Bn are close to diagonal matrices and a Levinson-type theorem can be applied to transform the sequence {M��n }��n=0 into a sequence of diagonal matrices. In the case that the Mn 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 k.