Consider the system with (a) Show that the matrix A has

Chapter 9, Problem 18E

(choose chapter or problem)

18. Consider the system with \(x^{\prime}(t)=A x(t), t \geq 0\) with

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

(a) Show that the matrix \(\text { A }\) has eigenvalues \(r_{1}=-1 \text { and } r_{2}=-3\) with corresponding eigenvectors \(u_{1}=\operatorname{col}(1,1)\)  and \(u_{2}=\operatorname{col}(1,1)\)

(b) Sketch the trajectory of the solution having initial vector \(x(0)=-u_{1}\)

(c) Sketch the trajectory of the solution having initial vector \(x(0)=-u_{2}\)

(d) Sketch the trajectory of the solution having initial vector \(x(0)=u_{2}-u_{1}\)

Equation Transcription:

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Text Transcription:

x'(t)=Ax(t), t \geq 0

A=[      1    -2    -2     1]

A      

r1=-1 and r2=-3      

u1=col (1, 1)      

u2=col (1,1)    

x(0)=-u1

x(0)=-u2

x(0)=u2-u1

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