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Following along the line of Exercise 11 in Section 6.3 and Exercise 15 in Section 7.2, suppose that a species of beetle has a life span of 4 years, and that a female in the first year has a survival rate of 1/2, in the second year a survival rate of 1/4, and in the third year a survival rate of 1/8 . Suppose additionally that a female gives birth, on the average, to two new females in the third year and to four new females in the fourth year. The matrix describing a single female s contribution in 1 year to the female population in the succeeding year is
where again the entry in the ith row and jth column denotes the probabilistic contribution that a female of age j makes on the next year s female population of age i.
a. Use the Geršgorin Circle Theorem to determine a region in the complex plane containing all the eigenvalues of A.
b. Use the Power method to determine the dominant eigenvalue of the matrix and its associated eigenvector.
c. Use Algorithm 9.4 to determine any remaining eigenvalues and eigenvectors of A.
d. Find the eigenvalues of A by using the characteristic polynomial of A and Newton s method. |
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