Q33E Find a 2脳2聽matrix聽聽such that... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q33E (page 324)

Find a $2 脳 2$ matrixsuch that
$\overset{鈫�}{X} (t) = [\begin{matrix} \begin{matrix} 2^{t} - & 6^{t} \end{matrix} \\ 2^{t} + 6^{t} \end{matrix}]$ is a trajectory of the dynamical systemrole="math" localid="1659527385729" $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$

Short Answer

Expert verified

So, the $2 脳 2$ matrix is $A = [\begin{matrix} 4 & - 2 \\ - 2 & 4 \end{matrix}]$ .

Step by step solution

Definition of trajectory of the dynamical system

A trajectory is the set of points in state space that are the future states resulting from a given initial state.

Given Data

Consider a dynamical system $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$ whose trajectory is $\overset{鈫�}{X} (t) = [\begin{matrix} 2^{t} - 6^{t} \\ 2^{t} + 6^{t} \end{matrix}]$

The objective is to find the matrix A.

The trajectory is written as,

$\overset{鈫�}{x} (t) = 2^{t} [\begin{matrix} 1 \\ 1 \end{matrix}] + 6^{t} [\begin{matrix} - 1 \\ 1 \end{matrix}]$

From here to fine a square matrix A of order 2 with Eigen vectors $[\begin{matrix} 1 \\ 1 \end{matrix}], [\begin{matrix} - 1 \\ 1 \end{matrix}]$ and corresponding Eigen values 2 and 6.

Find matrix

It is known that $\overset{鈫�}{A x} = 位 \overset{鈫�}{x}$ , so:

$A [\begin{matrix} 1 & - 1 \\ 1 & 1 \end{matrix}] = [\begin{matrix} 2 & - 6 \\ 2 & 6 \end{matrix}]$

And solve for A :

$A = [\begin{matrix} 2 & - 6 \\ 2 & 6 \end{matrix}] [\begin{matrix} 1 & - 1 \\ 1 & 1 \end{matrix}] = \frac{1}{2} [\begin{matrix} 2 & - 6 \\ 2 & 6 \end{matrix}] [\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}] = \frac{1}{2} [\begin{matrix} 8 & - 4 \\ - 4 & 8 \end{matrix}] = [\begin{matrix} 4 & - 2 \\ - 2 & 4 \end{matrix}]$

Hence, the matrix is $A = [\begin{matrix} 4 & - 2 \\ - 2 & 4 \end{matrix}]$ .

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91影视

Find a $2 脳 2$ matrixsuch that
$\overset{鈫�}{X} (t) = [\begin{matrix} \begin{matrix} 2^{t} - & 6^{t} \end{matrix} \\ 2^{t} + 6^{t} \end{matrix}]$ is a trajectory of the dynamical systemrole="math" localid="1659527385729" $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$

Short Answer

Step by step solution

Definition of trajectory of the dynamical system

Given Data

Find matrix

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Most popular questions from this chapter

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Logic and Functions

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91影视

Find a 2脳2matrixsuch thatX鈫�(t)=[2t-6t2t+6t]is a trajectory of the dynamical systemrole="math" localid="1659527385729" x鈫�(t+1)=Ax鈫�(t)

Short Answer

Step by step solution

Definition of trajectory of the dynamical system

Given Data

Find matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Applied Mathematics

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Find a $2 脳 2$ matrixsuch that
$\overset{鈫�}{X} (t) = [\begin{matrix} \begin{matrix} 2^{t} - & 6^{t} \end{matrix} \\ 2^{t} + 6^{t} \end{matrix}]$ is a trajectory of the dynamical systemrole="math" localid="1659527385729" $\overset{鈫�}{x} (t + 1) = A \overset{鈫�}{x} (t)$