91Ó°ÊÓ

Find an invertible matrix \(P\) and a matrix \(C\) of the form \(C=\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right]\) such that \(A=P C P^{-1}\). Sketch the first six points of the trajectory for the dynamical system \(\mathbf{x}_{k+1}=A \mathbf{x}_{k}\) with \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) and classify the origin as a spiral attractor, spiral repeller, or orbital center. $$A=\left[\begin{array}{rr} 0.1 & -0.2 \\ 0.1 & 0.3 \end{array}\right]$$

Find an invertible matrix \(P\) and a matrix \(C\) of the form \(C=\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right]\) such that \(A=P C P^{-1}\). Sketch the first six points of the trajectory for the dynamical system \(\mathbf{x}_{k+1}=A \mathbf{x}_{k}\) with \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) and classify the origin as a spiral attractor, spiral repeller, or orbital center. $$A=\left[\begin{array}{rr} 2 & 1 \\ -2 & 0 \end{array}\right]$$

Find an invertible matrix \(P\) and a matrix \(C\) of the form \(C=\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right]\) such that \(A=P C P^{-1}\). Sketch the first six points of the trajectory for the dynamical system \(\mathbf{x}_{k+1}=A \mathbf{x}_{k}\) with \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) and classify the origin as a spiral attractor, spiral repeller, or orbital center. $$A=\left[\begin{array}{rr} 1 & -1 \\ 1 & 0 \end{array}\right]$$

Find an invertible matrix \(P\) and a matrix \(C\) of the form \(C=\left[\begin{array}{rr}a & -b \\ b & a\end{array}\right]\) such that \(A=P C P^{-1}\). Sketch the first six points of the trajectory for the dynamical system \(\mathbf{x}_{k+1}=A \mathbf{x}_{k}\) with \(\mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) and classify the origin as a spiral attractor, spiral repeller, or orbital center. $$A=\left[\begin{array}{ll} 0 & -1 \\ 1 & \sqrt{3} \end{array}\right]$$