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Chapter 1: Q27E (page 1)

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,位2鈫�=0.9

Short Answer

Expert verified

So, the required solution is Atx0=伪蠀1+0.9t尾蠀2.

Step by step solution

01

Define the eigenvector

Eigenvector:An eigenvector ofAis a nonzero vectorvin Rnsuch that Av=位v, for some scalar位.

02

Note the given data

It is given that:

位1鈫�=1,位2鈫�=0.9

Given graph is:

03

Finding the required solution

We have:

A蠀1=蠀1A蠀2=0.9蠀2

Forx0=伪蠀1+尾蠀2 ,We have:

role="math" localid="1668079897383" Ax0=A(伪蠀1+尾蠀2)=伪A蠀1+尾A蠀2=伪蠀1+0.9尾蠀2

Therefore,Atx0=伪蠀1+0.9t尾蠀2.

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

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Most popular questions from this chapter

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, and suppose \(T\left( u \right) = {\mathop{\rm v}\nolimits} \). Show that \(T\left( { - u} \right) = - {\mathop{\rm v}\nolimits} \).

Give a geometric description of Span \(\left\{ {{v_1},{v_2}} \right\}\) for the vectors in Exercise 16.

Find the polynomial of degree 2[a polynomial of the form f(t)=a+bt+ct2] whose graph goes through the points localid="1659342678677" (1,-1),(2,3)and(3,13).Sketch the graph of the polynomial.

Consider the problem of determining whether the following system of equations is consistent for all \({b_1},{b_2},{b_3}\):

\(\begin{aligned}{c}{\bf{2}}{x_1} - {\bf{4}}{x_2} - {\bf{2}}{x_3} = {b_1}\\ - {\bf{5}}{x_1} + {x_2} + {x_3} = {b_2}\\{\bf{7}}{x_1} - {\bf{5}}{x_2} - {\bf{3}}{x_3} = {b_3}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of Span \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\). Then solve that problem.
  1. Define an appropriate matrix, and restate the problem using the phrase 鈥渃olumns of A.鈥�
  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation, and let Sand U be functions from \({\mathbb{R}^n}\) into \({\mathbb{R}^n}\) such that \(S\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) and \(\)\(U\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\). Show that \(U\left( v \right) = S\left( v \right)\) for all v in \({\mathbb{R}^n}\). This will show that Thas a unique inverse, as asserted in theorem 9. (Hint: Given any v in \({\mathbb{R}^n}\), we can write \({\mathop{\rm v}\nolimits} = T\left( {\mathop{\rm x}\nolimits} \right)\) for some x. Why? Compute \(S\left( {\mathop{\rm v}\nolimits} \right)\) and \(U\left( {\mathop{\rm v}\nolimits} \right)\)).
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