12. A stochastic matrix P is called doubly stochastic if Ei Pij = 1 for all j. It is called sub-stochastic if Ei Pij ≤ 1 for all j. Show that, if P is stochastic (respectively, doubly stochastic, sub-stochastic), then Pn is stochastic (respectively, doubly stochastic, sub-stochastic) for all n.

12. A stochastic matrix P is called doubly stochastic if Ei Pij = 1 for all j. It is called sub-stochastic if Ei Pij ≤ 1 for all j. Show that, if P is stochastic (respectively, doubly stochastic, sub-stochastic), then Pn is stochastic (respectively, doubly stochastic, sub-stochastic) for all n.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 14EQ

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Question

12. A stochastic matrix Pis called doubly stochastic if Σi Pij = 1 for all j. It is called sub-stochastic
if Ei Pij ≤ 1 for all j. Show that, if P is stochastic (respectively, doubly stochastic, sub-stochastic),
then Pn is stochastic (respectively, doubly stochastic, sub-stochastic) for all n.

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