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Chapter 7: Q7.6-35E (page 381)

(a). Consider a real n 脳 n matrix with n distinct real eigenvalues位1,鈥�,位nwhere|位i|猢�1for all i =1,鈥�,n. Letbe a trajectory of the dynamical systemx鈫�(t+1)=Ax鈫�(t). Show that this trajectory is bounded; that is, there is a positive number M such that鈭�x鈫�(t)鈭�猢�Mfor all positive integers t.

(b). Are all trajectories of the dynamical system

x鈫�(t+1)=(1101)x鈫�(t)

bounded? Explain.

Short Answer

Expert verified

(a).The trajectory has been provedM=鈭�i=1ncivi

(b). The all trajectories of this system are bounded.

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

Prove the trajectory (a):

Considervibe the corresponding eigenvector for the eigenvalue位i,i=1,鈥�,n. Thenv1,鈥�,vnis an eigen basis for鈩�n

We can write the equation asx(t)=鈭�i=1nci位itvi,

鈭�x(t)鈭�=鈭�i=1nci位tvi鈮�鈭�i=1n位itcivi鈮�鈭�i=1ncivi

Where we get,M=鈭�i=1ncivi

Hence proved.

03

Explain trajectories of the dynamical system (b):

The matrix A is upper triangular, so its eigenvalues are the entries on its main diagonal, which are位1,2=1.

Since,位1=位2=1猢�1.

Then, the all trajectories of this system are bounded

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Most popular questions from this chapter

A. Find the characteristic polynomial of the matrix

A=[010001abc]

B. Can you find a 3脳3matrix whose characteristic polynomial is-位2+17位2-5位+蟿蟿

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Orthogonal projection onto a line L inR3.
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