(Solved): 1. Let A=[3113]. Find the eigenvalues and eigenvectors of A, construct matrix P and diagonal ...

1. Let $$A=\left[\begin{array}{ll}3& 1\\ 1& 3\end{array}\right]$$. Find the eigenvalues and eigenvectors of $$A$$, construct matrix $$P$$ and diagonalized matrix $$B=P^{?1}AP$$. Solve linear systems $$\stackrel{?}{\mathbf{x}}=A\mathbf{x}$$ and the diagonalized system $$\stackrel{?}{\mathbf{y}}=B\mathbf{y}$$. Sketch phase portraits in both $$\mathbf{x}$$ and $$\mathbf{y}$$ plane. 2. Compute exponential of $$\left[\begin{array}{ll}0& i\\ ?i& 0\end{array}\right].$$ Use definition of $$\exp (A)$$. 3. Let $$A=\left[\begin{array}{rrr}1& 0& 0\\ 1& 2& 0\\ 1& 0& ?1\end{array}\right]$$ Find $$e^{At}$$ and solve the linear system $$\stackrel{?}{\mathbf{x}}=A\mathbf{x}$$. Find stable, unstable, and centre supspaces. Sketch phase portrait. 4. Solve $$\stackrel{?}{\mathbf{x}}=\left(\begin{array}{rr}0& 1\\ ?1& 2\end{array}\right)\mathbf{x}$$ 5. Show that in $${\mathbf{R}}^2$$ the only invariant lines for $$\stackrel{?}{\mathbf{x}}=A\mathbf{x},\mathbf{x}(0)={\mathbf{x}}_0$$ are the lines $$a{x}_1+b{x}_2=0$$, where $$\mathbf{v}=(?b,a{)}^T$$ is an eigenvector of $$A$$. 6. Solve $$\stackrel{?}{\mathbf{x}}=\left(\begin{array}{rrr}0& 2& 0\\ ?2& 0& 0\\ 2& 0& 6\end{array}\right)\mathbf{x}$$ Find stable, unstable, and centre supspaces. Sketch phase portrait.