Exercise 5.3.10

To calculate the vector after two stage, we could only multiply the matrix twice and multiply the vectors. To calculate the limit matrix we could just find the eigenvector corresponding to eigenvalue $1$ and use Theorem 5.20.

1.

The two-stage vectors and the fixed vectors are

$$\left(\begin{array}{c}\hfill 0.225\hfill \\ \hfill 0.441\hfill \\ \hfill 0.334\hfill \end{array}\right),\left(\begin{array}{c}\hfill 0.2\hfill \\ \hfill 0.6\hfill \\ \hfill 0.2\hfill \end{array}\right)$$

respectively.

2.

The two-stage vectors and the fixed vectors are

$$\left(\begin{array}{c}\hfill 0.375\hfill \\ \hfill 0.375\hfill \\ \hfill 0.25\hfill \end{array}\right),\left(\begin{array}{c}\hfill 0.4\hfill \\ \hfill 0.4\hfill \\ \hfill 0.2\hfill \end{array}\right)$$

respectively.

3.

The two-stage vectors and the fixed vectors are

$$\left(\begin{array}{c}\hfill 0.372\hfill \\ \hfill 0.225\hfill \\ \hfill 0.403\hfill \end{array}\right),\left(\begin{array}{c}\hfill 0.5\hfill \\ \hfill 0.2\hfill \\ \hfill 0.3\hfill \end{array}\right)$$

respectively.

4.

The two-stage vectors and the fixed vectors are

$$\left(\begin{array}{c}\hfill 0.252\hfill \\ \hfill 0.334\hfill \\ \hfill 0.414\hfill \end{array}\right),\left(\begin{array}{c}\hfill 0.25\hfill \\ \hfill 0.35\hfill \\ \hfill 0.4\hfill \end{array}\right)$$

respectively.

5.

The two-stage vectors and the fixed vectors are

$$\left(\begin{array}{c}\hfill 0.329\hfill \\ \hfill 0.334\hfill \\ \hfill 0.337\hfill \end{array}\right),\left(\begin{array}{c}\hfill \frac{1}{3}\hfill \\ \hfill \frac{1}{3}\hfill \\ \hfill \frac{1}{3}\hfill \end{array}\right)$$

respectively.

6.

The two-stage vectors and the fixed vectors are

$$\left(\begin{array}{c}\hfill 0.316\hfill \\ \hfill 0.428\hfill \\ \hfill 0.256\hfill \end{array}\right),\left(\begin{array}{c}\hfill 0.25\hfill \\ \hfill 0.5\hfill \\ \hfill 0.25\hfill \end{array}\right)$$

respectively.