Question 6.2.1: Find an eigenvector and associated eigenvalue for the matrix...

We apply the Power Method starting with x_0 = (1, 1, 1, 1, 1). Table 3 gives the vectors x _1, \ldots x _9.

The sequence settles to the vector (1.0000, 0.6667, −0.1667, 0.1111, 0.4444), and the sequence of scaling factors to 13.00. The components of the vector are recognizable as decimal approximations to rationals. Changing the decimals to equivalent rationals and multiplying by 18 to eliminate the fractions gives us

u=\left[\begin{array}{r}1.0000 \\0.6667 \\-0.1667 \\0.1111 \\0.4444\end{array}\right] \approx\left[\begin{array}{r}1 \\2 / 3 \\-1 / 6 \\1 / 9 \\4 / 9\end{array}\right] \Longrightarrow u=\left[\begin{array}{r}18 \\12 \\-3 \\2 \\8\end{array}\right]

To test our answer, we calculate

A u=\left[\begin{array}{rrrrr}245 & -254 & -252 & -46 & -224 \\161 & -168 & -174 & -32 & -148 \\-39 & 40 & 45 & 7 & 38 \\27 & -28 & -32 & -6 & -26 \\110 & -113 & -110 & -21 & -101\end{array}\right]\left[\begin{array}{r}18 \\12 \\-3 \\2 \\8\end{array}\right]=\left[\begin{array}{r}234 \\156 \\-39 \\26 \\104\end{array}\right]=13\left[\begin{array}{r}18 \\12 \\-3 \\2 \\8\end{array}\right]

confirming that u is an eigenvector and that λ = 13 is the associated eigenvalue.