Powers of Matrices

Abstract

In the previous chapter we presented known facts from the spectral theory of matrices in a coordinate-free way. We are, however, interested not simply in linear algebra, but mainly in the asymptotic behavior of dynamical systems, a central theme in this text.

We apply the knowledge we gained on the structure of linear operators on finite-dimensional vector spaces to investigate what happens to the sequence consisting of the powers of a matrix. Topics we cover include boundedness, convergence to zero, convergence, mean convergence (or Cesàro convergence), periodicity, and hyperbolic decomposition.