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Chapter 7: Q28E (page 324)

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

位1鈫�=1.2,位2鈫�=1.1

Short Answer

Expert verified

So, the required solution isAtx0=1.2t伪蠀1+1.1t尾蠀2.

Step by step solution

01

Define the eigenvector 

Eigenvector:An eigenvector of Ais a nonzero vectorvinRnsuch thatAv=位v, for some scalar位 .

02

Note the given data 

It is given that:

位1鈫�=1.2,位2鈫�=1.1

Given graph is:

03

Calculate the required matrix

We have:

A蠀1=1.2蠀1A蠀2=1.1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=1.2伪蠀1+1.1尾蠀2

Therefore,Atx0=1.2t伪蠀1+1.1t尾蠀2 .

Hence, the solution is Atx0=1.2t伪蠀1+1.1t尾蠀2.

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Most popular questions from this chapter

If a 2 脳 2 matrix A has two distinct eigenvalues位1and位2, show that A is diagonalizable.

(a) Give an example of a 3 脳 3 matrix A with as many nonzero entries as possible such that both span(e鈫�1) and span(e鈫�1,e鈫�2) are A-invariant subspaces of . See Exercise 65.

(b) Consider the linear space Vof all 3 脳 3 matrices A such that both span (e鈫�1) and span (e鈫�1,e鈫�2) are A-invariant subspaces of R3 . Describe the space V (the matrices in V 鈥渉ave a name鈥�), and determine the dimension of V.

Two interacting populations of coyotes and roadrunners can be modeled by the recursive equations

c(t + 1) = 0.75r(t)

r(t + 1) = 鈭�1.5c(t) + 2.25r(t).

For each of the initial populations given in parts (a) through (c), find closed formulas for c(t) and r(t).

(a)c(0)=100,r(0)=200(b)c(0)=r(0)=100(c)c(0)=500,r(0)=700

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

If v鈬赌 is an eigenvector of matrix A with associated eigenvalue 3 , show that v鈬赌is an image of matrix A .
See all solutions

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