Dorothy Goettler
NAE – HW 9
Problem 9.2.13
0
2
0
0.5
0
0
A
 0 0.25
0

0
0.125
0
4
0
0

0
a. As all terms along the diagonal are zero, it is known that all of the circles will be concentric with
centers at the origin. To determine the radius of the largest circle, each row of the matrix above is
examined to determine which one will yield the largest sum. In this case it is row 1. This indicates
that the eigenvalues of A are contained within a circle of radius 6, or  6.
b. The Power method is now used to determine the dominant eigenvalue of the matrix and its associated
eigenvector. Applying Algorithm 9.1 yielded the following results:
(133) = 0.69766854
x(133) = (1, 0.7166727, 0.2568099, 0.04601217)T
c. Algorithm 9.4 was used next to determine the remaining eigenvalues and eigenvectors of A.
Wielandt deflation is used to approximate the second most dominant eigenvalue and associated
eigenvector of the given matrix given an approximation to the initial eigenvalue and eigenvector.
d. The eigenvalues of A were also determined using the Newton-Raphson method.
The characteristic polynomial is P() = 4 - /4 – 1/16.
Eigenvalues: 1 = 0.6976684972,
2 = -0.2301775942 + 0.56965884i,
3 = -0.2301775942 - 0.56965884i,
4 = -0.237313308
e. The beetle population should approach zero.