AI Tutor
Back to Markov chains and Google's PageRank algorithm

Exercise 4

Avg difficulty 2.5

Consider the stochastic matrix A=[10.20.200.60.200.20.6]A = \left[\begin{array}{rrr} 1 & 0.2 & 0.2 \\ 0 & 0.6 & 0.2 \\ 0 & 0.2 & 0.6 \\ \end{array}\right].

  1. Find the eigenvalues of AA.

  2. Do the conditions of the Perron-Frobenius theorem apply to this matrix?

  3. Find the steady-state vectors of AA.

  4. What can we guarantee about the long-term behavior of a Markov chain defined by the matrix AA?

Austin, Understanding Linear Algebra, §4.5 Markov chains and Google's PageRank algorithm, Exercise 4

AI Tutor

Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.