Numerical Analysis/Power iteration exercises

Exercises

Consider the matrix, $A=\left[\begin{array}{c c c}6 & 2 & -1 \\2 & 5 & 1 \\-1 & 1 & 4 \end{array} \right]$ , and the vector, $\textbf{x}^{(0)} = \left[\begin{array}{c}1 \\1 \\ 1 \\\end{array} \right].$

By hand, do two iterations of the power method starting with $\textbf{x}^{(0)}.$

Solution, first iteration:

$A\textbf{x}^{(0)}=\left[ \begin{array}{c c c} 6&2&-1 \\ 2&5&1 \\ -1&1&4 \end{array} \right] \left[ \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right]=\left[ \begin{array}{c} 7 \\ 8 \\ 4 \end{array} \right],$ so the estimated eigenvalue is $\mu_1 = 8$ and the estimated eigenvector is $\textbf{x}^{(1)} =\frac{1}{8}A\textbf{x}^{(0)} = \left[ \begin{array}{c} \frac{7}{8} \\ 1 \\ \frac{1}{2} \end{array} \right]$ .

Solution, second iteration:

$A\textbf{x}^{(1)}=\left[ \begin{array}{c c c} 6&2&-1 \\ 2&5&1 \\ -1&1&4 \end{array} \right] \left[ \begin{array}{c} \frac{7}{8} \\ 1 \\ \frac{1}{2} \end{array} \right]=\left[ \begin{array}{c} \frac{27}{4} \\ \frac{29}{4} \\ \frac{17}{8} \end{array} \right],$ so the estimated eigenvalue is $\mu_2=\frac{29}{4}$ and the estimated eigenvector is $\textbf{x}^{(2)} =\frac{4}{29}A\textbf{x}^{(1)} = \left[ \begin{array}{c} \frac{27}{29} \\ 1 \\ \frac{17}{58} \end{array} \right]$ .

Use a computational software package to do 50 iterations. What was the result?

Solution, 50 iterations:

The estimated eigenvalue after 50 iterations is $\mu=7.5712$ with estimated eigenvector $\textbf{x}^{(50)}=\left[\begin{array}{c} 1.0000 \\ 0.7507 \\ -0.0698 \end{array} \right]$

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