Question 6.2.2: For the matrix A given in Example 1, find the eigenvalue (an...
For the matrix A given in Example 1, find the eigenvalue (and an associated eigenvector) farthest from λ_1 = 13.
To find λ_2, the eigenvalue of A farthest from λ_1 = 13, we first apply the Power Method to the matrix
B=A-13 I_5=\left[\begin{array}{rrrrr}232 & -254 & -252 & -46 & -224 \\161 & -181 & -174 & -32 & -148 \\-39 & 40 & 32 & 7 & 38 \\27 & -28 & -32 & -19 & -26 \\110 & -113 & -110 & -21 & -114\end{array}\right]
As in Example 1,we start with x_0 = (1, 1, 1, 1, 1). In this case, the convergence is slower, so we report only every 10th iteration in Table 4.
We see that s_k → −16, which implies that λ = −16 is an eigenvalue for B. Therefore λ_2 = −16 + 13 = −3 is the eigenvalue of A farthest from λ_1 = 13. Table 4 shows u = (1.000, 0.7142, −0.1429, 0.1429, 0.4286) is the associated eigenvector. We check this by computing
\begin{aligned}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}1.0000 \\0.7142 \\-0.1429 \\0.1429 \\0.4286\end{array}\right] \\& \approx\left[\begin{array}{r}-2.9758 \\-2.1266 \\0.4246 \\-0.4258 \\-1.2751\end{array}\right] \approx-3\left[\begin{array}{r}0.9919 \\0.7089 \\-0.1415 \\0.1419 \\0.4250\end{array}\right] \approx-3 u\end{aligned}
The approximations are a bit rough, but they can be refined by carrying more decimal places and computing additional iterations (see margin).
| k | x_k | s_k |
| 10 | (1.0000, 0.7033, −0.1483, 0.1483, 0.4450) | −15.76 |
| 20 | (1.0000, 0.7078, −0.1461, 0.1461, 0.4383) | −15.86 |
| 30 | (1.0000, 0.7107, −0.1447, 0.1447, 0.4340) | −15.92 |
| 40 | (1.0000, 0.7123, −0.1438, 0.1438, 0.4315) | −15.96 |
| 50 | (1.0000, 0.7132, −0.1434, 0.1434, 0.4302) | −15.98 |
| 60 | (1.0000, 0.7137, −0.1431, 0.1431, 0.4294) | −15.99 |
| 70 | (1.0000, 0.7140, −0.1430, 0.1430, 0.4290) | −15.99 |
| 80 | (1.0000, 0.7141, −0.1429, 0.1429, 0.4288) | −16.00 |
| 90 | (1.0000, 0.7142, −0.1429, 0.1429, 0.4287) | −16.00 |
| 100 | (1.0000, 0.7142, −0.1429, 0.1429, 0.4286) | −16.00 |
| Table 4 The Shifted PowerMethod Applied to B = A − 13I_5 | ||