(a) Show that the matrix has the repeated eigenvalue r = 1

Chapter 9, Problem 39E

(choose chapter or problem)

39. (a) Show that the matrix

\(\mathbf{A}=\left[\begin{array}{rrr} 2 & 1 & 1 \\ 1 & 2 & 1 \\ -2 & -2 & -1 \end{array}\right]\)

has the repeated eigenvalue r  = 1 of multiplicity 3 and that all the eigenvectors of A are of the

form \(u=s \operatorname{col}(-1,1,0)+v \operatorname{col}(-1,0,1)\)

(b) Use the result of part (a) to obtain two linearly independent solutions to the system \(x^{\prime}=A x\) of the form

\(x_{1}(t)=e^{t} u_{1} \text { and } x_{2}(t)=e^{t} u_{2}\)

(c) To obtain a third linearly independent solution to \(x^{\prime}=A x, \text { try } x_{3}(t)=t e^{t} u_{3}+e^{t} u_{4}\)   [Hint: Show that \(u_{3} \text { and } u_{4}\)  must satisfy

\((A-1) u_{3}=0,(A-1) u_{4}=u_{3}\)

Choose \(u_{3}\) an eigenvector of A, so that you can solve for \(u_{4}\).

(d) What is \((A-1)^{2} u_{4}\)?

Equation Transcription:

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Text Transcription:

A=[2 & 1 & 1 \\1 & 2 & 1 \\-2 & -2 & -1

u=s col(-1,1,0)+v col(-1,0,1)

x^\prime=A x

x_1(t)=e^t u_1 and  x_2(t)=e^t u_2

x^\prime=A x, try x_3(t)=t e^t u_3+e^t u_4

u_3 and  u_4

(A-1) u_3=0,(A-1) u_4=u_3

u_3

u_4

(A-1)^2 u_4