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Chapter 7: Q22E (page 372)

Find all complex eigenvalues of the matrices in Exercises 20 through 26 (including the real ones, of course). Do not use technology. Show all your work.

22)[13-410]

Short Answer

Expert verified

位1,2=11卤i332

Step by step solution

01

Define eigen values

The scalar values correlated to the system of linear equations, mostly comprising of matrix are called eigen values.

02

Find complex eigenvalues of the matrices

If the n 脳 n matrix A has real entries, its complex eigenvalues will always occur in complex conjugate pairs.

Solve,

det(A-位濒)=01-位3-410-位=01-位10-位+12=0位2+11位+22=0位1,2=11卤121-882=11卤332

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Most popular questions from this chapter

Find all2脳2matrices for whiche鈫�1=(10)is an eigenvector with associated eigenvalue 5.

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]

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.

[2-2001-100003-4002-3]

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.

Consider the matrix A=|abba|where aand bare arbitrary constants. Find all eigenvalues of A.
See all solutions

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