Partial Norms and the Convergence of General Products of Matrices

Motivated by the theory of inhomogeneous Markov chains, we determine a sufficient condition for the convergence to 0 of a general product formed from a sequence of real or complex matrices. When the matrices have a common invariant subspace H, we give a sufficient condition for the convergence to 0 on H of a general product. Our result is applied to obtain a condition for the weak ergodicity of an inhomogeneous Markov chain. We compare various types of contractions which may be defined for a single matrix, such as paracontraction, l– contraction, and H–contraction, where H is an invariant subspace of the matrix. Work supported by NSF Grant DMS-9306357. Research supported in part by NSF Grant DMS-9424346. This research was partly performed during a visit by this author to the University of Bielefeld, Germany, supported by the Sonderforschungsbereich “Diskrete Strukturen in der Mathematik”, Universität Bielefeld.