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Chapter 7: Q47E (page 325)
If is an eigenvector of matrix A, show that is in the image of A.or in the kernel ofA.
It is shown that is in the image of A .
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For an arbitrary positive integer n, give a matrix A without real eigenvalues.
25: Consider a positive transition matrix
meaning that a, b, c, and dare positive numbers such that a+ c= b+ d= 1. (The matrix in Exercise 24 has this form.) Verify that
and
are eigenvectors of A. What are the associated eigenvalues? Is the absolute value of these eigenvalues more or less than 1?
Sketch a phase portrait.
Consider an matrix such that the sum of the entries in each row is . Show that the vector
In is an eigenvector of A. What is the corresponding eigenvalue?
Show that similar matrices have the same eigenvalues. Hint: Ifis an eigenvector of, thenrole="math" localid="1659529994406" is an eigenvector of A.
28 : Consider the isolated Swiss town of Andelfingen, inhabited by 1,200 families. Each family takes a weekly shopping trip to the only grocery store in town, run by Mr. and Mrs. Wipf, until the day when a new, fancier (and cheaper) chain store, Migros, opens its doors. It is not expected that everybody will immediately run to the new store, but we do anticipate that 20% of those shopping at Wipf鈥檚 each week switch to Migros the following week. Some people who do switch miss the personal service (and the gossip) and switch back: We expect that 10% of those shopping at Migros each week go to Wipf鈥檚 the following week. The state of this town (as far as grocery shopping is concerned) can be represented by the vector
where w(t) and m(t) are the numbers of families shopping at Wipf鈥檚 and at Migros, respectively, t weeks after Migros opens. Suppose w(0) = 1,200 and m(0) = 0.
a. Find a 2 脳 2 matrix A such that role="math" localid="1659586084144" . Verify that A is a positive transition matrix. See Exercise 25.
b. How many families will shop at each store after t weeks? Give closed formulas. c. The Wipfs expect that they must close down when they have less than 250 customers a week. When does that happen?
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