Power and exponential of the following Jordan form

Question

Let

$$J= \left(\begin{matrix} 4 & 1 & 0 \\ 0 & 4 & 1 \\ 0 & 0 & 4 \\ \end{matrix}\right)$$

What $e^{Jt}$ and $J^t$ look like? Can you help me, please?

Answer 1

Hint: Note that $J = 4I + N$, where $$ N = \pmatrix{0&1&0\\0&0&1\\0&0&0}. $$ We have $\exp(Jt) = \exp((4I + N)t) = \exp(4It)\exp(Nt)$.

Similarly, we can expand $(4I + N)^t$ using the binomial theorem for any integer $t$.

Both of these are made easier to compute by the fact that $N^3 = 0$.