###### Activity4.5.4

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.

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