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Partial norms and the convergence of general products of matrices

Michael Neumann·Hans Schneider
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Abstract

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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Michael Neumann, Hans Schneider. 1998-02-23. Partial norms and the convergence of general products of matrices. https://arxiv.org/abs/math/9802108

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