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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