For the matrix A given in Example 1, find the eigenvalue (and an associated eigenvector) farthest from λ1 = 13.

Question

For the matrix A given in Example 1, find the eigenvalue (and an associated eigenvector) farthest from λ[latex]_1[/latex] = 13.

Accepted Answer

To find λ[latex]_2[/latex], the eigenvalue of A farthest from λ[latex]_1[/latex] = 13, we first apply the Power Method to the matrix

[latex]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} ight][/latex]

As in Example 1,we start with x[latex]_0[/latex] = (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[latex]_k[/latex] → −16, which implies that λ = −16 is an eigenvalue for B. Therefore λ[latex]_2[/latex] = −16 + 13 = −3 is the eigenvalue of A farthest from λ[latex]_1[/latex] = 13. Table 4 shows u = (1.000, 0.7142, −0.1429, 0.1429, 0.4286) is the associated eigenvector. We check this by computing

[latex]\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} ight]\left[\begin{array}{r}1.0000 \0.7142 \-0.1429 \0.1429 \0.4286\end{array} ight] \& \approx\left[\begin{array}{r}-2.9758 \-2.1266 \0.4246 \-0.4258 \-1.2751\end{array} ight] \approx-3\left[\begin{array}{r}0.9919 \0.7089 \-0.1415 \0.1419 \0.4250\end{array} ight] \approx-3 u\end{aligned}[/latex]

The approximations are a bit rough, but they can be refined by carrying more decimal places and computing additional iterations (see margin).

Question 6.2.2: For the matrix A given in Example 1, find the eigenvalue (an...

Related Answered Questions

For the matrix A given in Example 1, find the eigenvalue that is closest to c = 4. ...

Verified Answer:

For the matrix A in Example 1, find the smallest eigenvalue (in absolute value) and an associated eigenvector. ...

Verified Answer:

Find an eigenvector and associated eigenvalue for the matrix A =[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](3) ...

Verified Answer:

Show that λ = 0 is an eigenvalue for the matrix A =[3 -1 5 2 1 0 4 1 2] ...

Verified Answer:

Find the eigenvalues and a basis for each eigenspace of A =[0 2 -1 1 -1 0 1 -2 0] ...

Verified Answer:

Find the eigenvalues and a basis for each eigenspace of A =[1 -2 1 -1 0 1 -1 -2 3] ...

Verified Answer:

Find the eigenvalues and a basis for each eigenspace of A =[1 -3 3 2 -2 2 2 0 0] ...

Verified Answer:

Find the eigenvalues and a basis for each eigenspace for the matrix A = [2 -1 -1 2] ...

Verified Answer:

Find the eigenvalues for A =[3 3 6 -4]. ...

Verified Answer:

Take as known that λ = 3 and λ = 2 are eigenvalues for A =[4 4 -2 1 4 -1 3 6 -1]Find the eigenvectors associated with each eigenvalue. ...

Verified Answer: