Partial norms and the convergence of general products of matrices

Details Partial norms and the convergence of general products of matrices Partial norms and the convergence of general products of matrices

1998-02-23

arxiv.org/abs/math/9802108v1

Mathematics

Rings and Algebras

Scientific paper

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.

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