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Chapter 7: Q7.6-33E (page 381)

Consider a real 2 脳 2 matrix A with eigenvalues p卤iqand corresponding eigenvectorsv鈫�卤iw鈫�.Show that if a real vectorx鈫�0is written asx鈫�0=c1(v鈫�+iw鈫�)+c2(v鈫�-iw鈫�)then c2=c炉1.

Short Answer

Expert verified

The solution concluded asc1=c2炉andc2=c1炉

Step by step solution

01

Define eigenvalue:

Eigenvalues are a set of specialized scales associated with a system of linear equations. The corresponding eigenvalue, often denoted by 位.

02

To show the real vector:

As givenx0is a real vector,

x0=x0炉=c1炉(v-iw)+c2炉(v+iw)

So from this we can conclude

c1=c2炉andc2=c1炉

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Most popular questions from this chapter

Consider ann脳n matrix such that the sum of the entries in each row is . Show that the vector

e鈫�=11..1

InRn is an eigenvector of A. What is the corresponding eigenvalue?

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 eigensspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.

For each of the matrices A in Exercise1 through20,find all (real)eigenvalues.Then find a basis of each eigenspaces,and diagonalize A, if you can. Do not use technology.

[1111]

find an eigenbasis for the given matrice and diagonalize:

A=114[123246369]

consider the dynamical system

x鈫�(t+1)=[1.100位]x鈫�(t).

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

位=1.2
See all solutions

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