Activity 4.5.4

We will explore the meaning of the Perron-Frobenius theorem in this activity.

1. Consider the matrix \(C = \left[\begin{array}{rr} 0 \amp 0.5 \\ 1 \amp 0.5 \\ \end{array}\right] \text{.}\) This is a positive matrix, as we saw in the previous example. Find the eigenvectors of \(C\) and verify there is a unique steady-state vector.

2. Using the Sage cell below, construct the Markov chain with initial vector \(\xvec_0= \twovec{1}{0}\) and describe what happens to \(\xvec_k\) as \(k\) becomes large.

3. Construct another Markov chain with initial vector \(\xvec_0=\twovec{0.2}{0.8}\) and describe what happens to \(\xvec_k\) as \(k\) becomes large.

4. Consider the matrix \(D = \left[\begin{array}{rrr} 0 \amp 0.5 \amp 0 \\ 1 \amp 0.5 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array}\right]\) and compute several powers of \(D\) below.

   Determine whether \(D\) is a positive matrix.

5. Find the eigenvalues of \(D\) and then find the steady-state vectors. Is there a unique steady-state vector?

6. What happens to the Markov chain defined by \(D\) with initial vector \(\xvec_0 =\threevec{1}{0}{0}\text{?}\) What happens to the Markov chain with initial vector \(\xvec_0=\threevec{0}{0}{1}\text{.}\)

7. Explain how the matrices \(C\) and \(D\text{,}\) which we have considered in this activity, relate to the Perron-Frobenius theorem.