Explanation Given: The given matrix A is, A = [ − 1 − 4 − 2 − 4 − 5 − 6 4 8 7 ] Approach: Solve the vector differential equation x ' = A x in the case where A is a defective matrix by constructing cycles of generalized eigenvectors and forming corresponding solutions x k ( t ) as, Let, A be an n × n matrix. Suppose the eigenvalue λ has multiplicity m and that the dimension of the corresponding eigenspace is r ≤ m . Then there exist m linearly independent solutions to x ' = A x that can be constructed in r cycles (one for each i with 1 ≤ i ≤ r ) of the form. x 0 ( i ) ( t ) = e λ t v 0 ( i ) , x 1 ( i ) ( t ) = e λ t ( v 1 ( i ) + t v 0 ( i ) ) x 2 ( i ) ( t ) = e λ t ( v 2 ( i ) + t v 1 ( i ) + 1 2 ! t 2 v 0 ( i ) ) ⋮ x k i ( i ) ( t ) = e λ t ( v k i ( i ) ) + t v k i − 1 ( i ) + ⋯ + 1 k ! t k i v 0 ( i ) Where each k i ≥ 0 , and k 1 + k 2 + ⋯ + k r = m − r . Calculation: Consider the given matrix A , A = [ − 1 − 4 − 2 − 4 − 5 − 6 4 8 7 ] The characteristic equation of A is given as det ( A − λ I ) = 0 , det ( A − λ I ) = 0 | [ − 1 − 4 − 2 − 4 − 5 − 6 4 8 7 ] − [ λ 0 0 0 λ 0 0 0 λ ] | = 0 | [ − 1 − λ − 4 − 2 − 4 − 5 − λ − 6 4 8 7 − λ ] | = 0 Solve the above equation for the eigenvalues ( λ ) of A , | − 1 − λ − 4 − 2 − 4 − 5 − λ − 6 4 8 7 − λ | = 0 λ 3 − λ 2 + 3 λ + 5 = 0 λ = − 1 , 1 + 2 i , 1 − 2 i Therefore, the eigenvalues of A are − 1 , 1 + 2 i , 1 − 2 i . The eigenvector corresponding to the eigenvalue λ 1 = − 1 is determined based on the linear system given as, ( A − λ I ) v = 0 ( [ − 1 − 4 − 2 − 4 − 5 − 6 4 8 7 ] − [ λ 0 0 0 λ 0 0 0 λ ] ) [ v 1 v 2 v 3 ] = [ 0 0 0 ] [ − 1 − λ − 4 − 2 − 4 − 5 − λ − 6 4 8 7 − λ ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] Substitute − 1 for λ in the above equation. [ − 1 − ( − 1 ) − 4 − 2 − 4 − 5 − ( − 1 ) − 6 4 8 7 − ( − 1 ) ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] [ − 1 + 1 − 4 − 2 − 4 − 5 + 1 − 6 4 8 7 + 1 ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] [ 0 − 4 − 2 − 4 − 4 − 6 4 8 8 ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] The augmented matrix of the above matrix is, [ 0 − 4 − 2 − 4 − 4 − 6 4 8 8 | 0 0 0 ] The reduced row-echelon form is, [ 1 0 1 0 1 1 2 0 0 0 | 0 0 0 ] Thus, [ 1 0 1 0 1 1 2 0 0 0 ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] [ v 1 + v 3 v 2 + 1 2 v 3 0 ] = [ 0 0 0 ] v 1 + v 3 = 0 v 2 + 1 2 v 3 = 0 The equations obtained are, v 1 + v 3 = 0 v 1 = − v 3 And, v 2 + 1 2 v 3 = 0 v 2 = − 1 2 v 3 For v 3 = 2 r then v 1 = − 2 r and v 2 = − r as v 2 is the independent variable. Thus, v 1 = − 2 r , v 2 = − r and v 3 = 2 r , the solution to this system is, v = [ v 1 , v 2 , v 3 ] T = [ − 2 r , − r , 2 r ] T = [ − 2 r − r 2 r ] : r ∈ ℝ = r [ − 2 − 1 2 ] : r ∈ ℝ The eigenvector corresponding to λ 1 = − 1 is v 1 = [ − 2 − 1 2 ] . Thus, x 1 ( t ) = e − t [ − 2 − 1 2 ] . The eigenvector corresponding to the eigenvalue λ 2 = 1 + 2 i is determined based on the linear system given as, ( A − λ I ) v = 0 ( [ − 1 − 4 − 2 − 4 − 5 − 6 4 8 7 ] − [ λ 0 0 0 λ 0 0 0 λ ] ) [ v 1 v 2 v 3 ] = [ 0 0 0 ] [ − 1 − λ − 4 − 2 − 4 − 5 − λ − 6 4 8 7 − λ ] [ v 1 v 2 v 3 ] = [ 0 0 0 ] Substitute 1 + 2 i for λ in the above equation

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 9.11, Problem 19AP

To determine

The general solution to the system x'=Ax for the matrix A=[−1−4−2−4−5−6487].

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.11, Problem 18AP

Chapter 9.11, Problem 20AP

Students have asked these similar questions

dent Application X GA spinner is divided into five cox | + 9/26583471/4081d162951bfdf39e254aa2151384b7 A spinner is divided into five colored sections that are not of equal size: red, blue, green, yellow, and purple. The spinner is spun several times, and the results are recorded below: Spinner Results Color Frequency Red 5 Blue 11 Green 18 Yellow 5 Purple 7 Based on these results, express the probability that the next spin will land on purple as a fraction in simplest form. Answer Attempt 1 out of 2 Submit Answer 0 Feb 12 10:11 O

2 5x + 2–49 2 x+10x+21

5x 2x+y+ 3x + 3y 4 6

Chapter 9 Solutions

Differential Equations and Linear Algebra (4th Edition)

Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 5TFR Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 7TFR Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 9TFR Ch. 9.1 - Prob. 10TFR

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

The general solution to the system x ' = A x for the matrix A = [ − 1 − 4 − 2 − 4 − 5 − 6 4 8 7 ] .

Solution Summary: The author explains how to solve the vector differential equation x'=Ax in the case where A is a defective matrix.

Concept explainers

Videos

The general solution to the system x'=Ax for the matrix A=[−1−4−2−4−5−6487].

Want to see the full answer?

Chapter 9 Solutions