List the five upper Jordan canonical forms for a $4\times 4$ matrix $A$ with a real eigenvalue $\lambda$ of multiplicity $4$ and give the corresponding geometric multiplicities in each case. What is the form of the solution of the intial value problem $\bf{\dot{x}}=Ax$
Solution so far
The five upper Jordan canonical forms are $$ \left[ {\begin{array}{cccc} \lambda & 1 & 0 & 0 \\ 0 & \lambda & 1 & 0 \\ 0 & 0 &\lambda & 1 \\ 0 & 0 & 0 & \lambda \end{array} } \right] $$ $$ \begin{bmatrix} \lambda & 0 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda \end{bmatrix} \quad \begin{bmatrix} \lambda & 1 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda \end{bmatrix} \quad \begin{bmatrix} \lambda & 1 & 0 & 0 \\ 0 & \lambda & 1 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda \end{bmatrix} \quad \begin{bmatrix} \lambda & 1 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & \lambda \end{bmatrix} $$ But I am having issue finding the solution of the ivp in each case. I am not sure what the matrix A is in each case.
The solution is $x(t)=e^{tA}x_0$. For a Jordan block such as $$ J=\left[\begin{array}{cccc}\lambda & 1 & 0 & 0 \\ 0 & \lambda & 1 & 0 \\ 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & \lambda\end{array} \right] = \lambda I+\left[\begin{array}{cccc}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{array}\right] = \lambda I + N, $$ the matrices $I$ and $N$ commute, which allows you to distribute the sum over the exponential: \begin{align} e^{tJ} & = e^{t\lambda I}e^{tN} \\ & = e^{\lambda t}e^{tN} \\ & = e^{\lambda t}\left[I+tN+\frac{1}{2!}t^2N^2+\frac{1}{3!}t^3N^3\right]. \end{align} The series terminates at $N^3$ because $N^4=0$. You get $$ e^{tA} = e^{\lambda t}\left[\begin{array}{cccc} 1 & t & \frac{t^2}{2!} & \frac{t^3}{3!} \\ 0 & 1 & t & \frac{t^2}{2!} \\ 0 & 0 & 1 & t \\ 0 & 0 & 0 & 1 \end{array}\right] $$ The only part missing is the transition matrix $A = V^{-1}JV$ where $J$ is in Jordan canonical form. $e^{tA} = V^{-1}e^{tJ}V$, and $e^{tJ}$ consists of blocks of the form shown above.