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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