##### Activity4.5.2

Suppose you live in a country with three political parties $$P\text{,}$$ $$Q\text{,}$$ and $$R\text{.}$$ We use $$P_k\text{,}$$ $$Q_k\text{,}$$ and $$R_k$$ to denote the percentage of voters voting for that party in election $$k\text{.}$$

Voters will change parties from one election to the next as shown in the figure. We see that 60% of voters stay with the same party. However, 40% of those who vote for party $$P$$ will vote for party $$Q$$ in the next election.

1. Write expressions for $$P_{k+1}\text{,}$$ $$Q_{k+1}\text{,}$$ and $$R_{k+1}$$ in terms of $$P_k\text{,}$$ $$Q_k\text{,}$$ and $$R_k\text{.}$$

2. If we write $$\xvec_k = \threevec{P_k}{Q_k}{R_k}\text{,}$$ find the matrix $$A$$ such that $$\xvec_{k+1} = A\xvec_k\text{.}$$

3. Explain why $$A$$ is a stochastic matrix.

4. Suppose that initially 40% of citizens vote for party $$P\text{,}$$ 30% vote for party $$Q\text{,}$$ and 30% vote for party $$R\text{.}$$ Form the vector $$\xvec_0$$ and explain why $$\xvec_0$$ is a probability vector.

5. Find $$\xvec_1\text{,}$$ the percentages who vote for the three parties in the next election. Verify that $$\xvec_1$$ is also a probabilty vector and explain why $$\xvec_k$$ will be a probability vector for every $$k\text{.}$$

6. Find the eigenvalues of the matrix $$A$$ and explain why the eigenspace $$E_1$$ is a one-dimensional subspace of $$\real^3\text{.}$$ Then verify that $$\vvec=\threevec{1}{2}{2}$$ is a basis vector for $$E_1\text{.}$$

7. As every vector in $$E_1$$ is a scalar multiple of $$\vvec\text{,}$$ find a probability vector in $$E_1$$ and explain why it is the only probability vector in $$E_1\text{.}$$

8. Describe what happens to $$\xvec_k$$ after a very long time.

in-context