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Chapter 7: Q31E (page 324)

Consider the dynamical system

X鈫�(t+1)=[1.100鈪�]x鈫�(t).

Sketch a phase portrait of this system for the given values of 位:

位=1

Short Answer

Expert verified

The sketch a phase portrait of this system for the given values of is shown as below:

Step by step solution

01

Definition of matrix

A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

02

Sketch the phase portrait

Consider the given matrix,

A=1.1001

Clearly,is diagonal, so for any:

X0=X1X2

, we have:

AtX0=(1.1)t001X1X2=(1.1)tX1X2

So, the required sketch is shown as below using the technology.

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Most popular questions from this chapter

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[5-421]

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?

27: a. Based on your answers in Exercises 24 and 25, find closed formulas for the components of the dynamical system

x炉(t+1)=[0.50.250.50.75]x炉(t)

with initial value x0鈫�=e1鈫�. Then do the same for the initial value x0鈫�=e2鈫�. Sketch the two trajectories.

b. Consider the matrix

A=[0.50.250.50.75]

.

Using technology, compute some powers of the matrix A, say, A2, A5, A10, . . . .What do you observe? Diagonalize matrix Ato prove your conjecture. (Do not use Theorem 2.3.11, which we have not proven

yet.)

c. If A=[abcd]

is an arbitrary positive transition matrix, what can you say about the powers Atas t goes to infinity? Your result proves Theorem 2.3.11c for the special case of a positive transition matrix of size 2 脳 2.

For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

5.(45-2-2)

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[-3040-10-273]
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