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Chapter 7: Q26E (page 324)

Consider a dynamical system x鈫�(t+1)=Ax鈫�(t)with two components. The accompanying sketch shows the initial state vectorlocalid="1668400938891" x鈫�0and two eigen vectorslocalid="1668400903162" 蠀1鈫�鈥�鈥�and鈥�鈥�蠀2鈫�of A (with eigen valueslocalid="1668400914784" 位1鈫�and位2鈫�respectively). For the given values oflocalid="1668400926197" 位1鈫�and位2鈫�, draw a rough trajectory. Consider the future and the past of the system.

位1鈫�=1.1,位2鈫�=1

Short Answer

Expert verified

So, the required solution isAtx0=1.1t伪蠀1+0.9t尾蠀2.

Step by step solution

01

Define the eigenvector

Eigenvector: An eigenvector ofA is a nonzero vector vin role="math" localid="1668401193807" Rnsuch that role="math" localid="1668401247797" Av=位惫, for some scalar 位.

02

Note the given data

It is given that:

位1鈫�=1.1,位2鈫�=1

Given graph is:

03

Finding the required solution

We have:

A蠀1=1.1蠀1A蠀2=蠀2

Forx0=伪蠀1+尾蠀2,We have:

Ax0=A(伪蠀1+尾蠀2)=伪A蠀1+尾A蠀2=1.1伪蠀1+尾蠀2

Therefore,Atx0=1.1t伪蠀1+0.9t尾蠀2.

Hence, the solution is Atx0=1.1t伪蠀1+0.9t尾蠀2.

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Most popular questions from this chapter

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

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Representing the reflection about the planex-2y+2z=0.
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