Given the matrix: $A=\begin{pmatrix}-2&0&0\\4&-2&0\\1&0&-2\end{pmatrix}$, find $e^{At}$.
I found the eigenvalues to be $-2,-2,-2$. I need to use the Jordan form to solve it. I'm practicing for exam in a couple of hrs. Thank you. I found the eigenvectors, $v_1=(0,1,0), v_2=(0,1,1), V_3=(1,0,0).$
I think the Jordan form is $e^{Jt}=\begin{pmatrix}e^{-2t}&1&0\\0&e^{-2t}&1\\0&0&e^{-2t}\end{pmatrix}$,Is this right?
Next I need to calculate $Ve^{Jt}V^{-1}$ Actually I do know the answer Im just practicing the way to do it. Can anybody tell me whats wrong with V_3? I found that $(A+2I)^3=0$ and I thought any eigenvector satisfies $0\cdot v_3=0$?!