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Chapter 7: Q29E (page 355)

For the matrices A and the vectorsx鈫�0in Exercises 25 through 29, findlimt鈫�鈭�(Atx鈫�0). Feel free to use Theorem 7.4.1.

A=[0.50.20.20.20.30.50.30.50.3],x0鈫�=[0.50.20.3]

Short Answer

Expert verified

The equilibrium distributionlimt鈫�鈭�Atxo=2729843184

Step by step solution

01

Define Matrix

A matrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns.

02

Finding the equilibrium distribution xequ of A

Ax=x[0.50.20.20.20.30.50.30.50.3]x1x2x3=x1x2x30.5x1+0.2x2+0.2x3=x1,0.2x1+0.3x2+0.5x3=x2,0.3x1+0.5x2+0.3x3=x3,x1+x2+x3=1x1=27x2=2984x3=3184xequ=2729843184

03

Finding the Value of limt→∞(Atx0→)

limt鈫�鈭�Atx鈫�0=2729843184

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Most popular questions from this chapter

22: Consider an arbitrary n 脳 n matrix A. What is the relationship between the characteristic polynomials of A and AT ? What does your answer tell you about the eigenvalues of A and AT ?

Show that similar matrices have the same eigenvalues. Hint:v鈫� Ifis an eigenvector ofS-1AS, thenrole="math" localid="1659529994406" Sv鈫�is an eigenvector of A.

Consider the matrix

A=[344-3]

a. Use the geometric interpretation of this transformation as a reflection combined with scaling to find the eigenvaluesA.

b. Find an eigen basis for A.

c. Diagonalize 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

x炉(t)=[wtm(t]]

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" x炉(t++1)=Ax鈫�(t). 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?

Is v鈬赌an eigenvector ofA-1? If so, what is the eigenvalue?
See all solutions

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