41 Reducing to Jordan blocks

From the Jordan canonical form, we have

\exp(tA)=S\begin{bmatrix}\exp(tJ_{k_{1}}(\lambda_{1}))&0&\dots&0\\ 0&\exp(tJ_{k_{2}}(\lambda_{2}))&0&\dots\\ \vdots&0&\ddots&0\\ 0&\dots&0&\exp(tJ_{k_{r}}(\lambda_{r}))\end{bmatrix}S^{-1},

so we consider a typical block $J_{k}(\lambda)$ . Now

J_{k}(\lambda)=\begin{bmatrix}\lambda&0&0&\dots&0\\ 0&\lambda&0&0&\dots\\ \vdots&0&\ddots&\ddots&0\\ \vdots&0&0&\ddots&0\\ 0&0&\dots&0&\lambda\end{bmatrix}+\begin{bmatrix}0&1&0&\dots&0\\ 0&0&1&0&\dots\\ \vdots&0&\ddots&\ddots&0\\ \vdots&0&0&\ddots&1\\ 0&0&\dots&0&0\end{bmatrix}

which we write as