To find a general solution to the system proceed as

Chapter 9, Problem 47E

(choose chapter or problem)

47. To find a general solution to the system

\(\mathbf{x}^{\prime}=\mathbf{A x}=\left[\begin{array}{rrr}  1 & 3 & -1 \\  3 & 0 & 1 \\

-1 & 1 & 2  \end{array}\right] \mathbf{x}\)

proceed as follows:

(a) Use a numerical root-finding procedure to approximate the eigenvalues.

(b) If r is an eigenvalue, then let \(u=\operatorname{col}\left(u_{1}, u_{2}, u_{3}\right)\) be an eigenvector associated with r. To solve for \(u\), assume \(u_{1}=1\). (If not \(u_{1}\) then either \(u_{2} \text { or } u_{3}\) may be chosen to be 1. Why?) Now solve the system

\(\left(\begin{array}{l}  A-r I \\  \end{array}\right)\left[\begin{array}{l}  1 \\  u_{2} \\  u_{3}  \end{array}\right]=\left[\begin{array}{l}  0 \\  0 \\  0  \end{array}\right]\)

for \(u_{2} \text { and } u_{3}\). Use this procedure to find approximations for three linearly independent eigenvectors for \(A\)

(c) Use these approximations to give a general solution to the system.

Equation Transcription:

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Text Transcription:

x'=Ax=[ -1   1    2   3      0      1   1      3    -1 ]x

u=col(u1,u2,u3)    

u  

u1=1    

u1  

u2 or u3    

(A-rI)[ u3 u2 1 ]=[ 000 ]

u2 and u3

A

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