Matrix exponential without series expansion

Dear **MHF** members,

my problem is explained below.

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**Problem**. Compute the matrix exponential of the matrix

$\displaystyle A:=

\left(

\begin{array}{rrrr}

1 & 0 & -1 & 1 \\

0 & 1 & 1 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 1 & 0 \\

\end{array}

\right).$

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I can compute the matrix exponential by using the series expansion

$\displaystyle \mathrm{e}^{At}=\mathrm{I}+\displaystyle\sum_{\ell =1}^{\infty}\displaystyle\frac{1}{\ell!}A^{\ell}t^ {\ell},$

where

$\displaystyle A^{n}=

\left(

\begin{array}{rrrr}

1 & 0 & -1 & 1 \\

0 & 1 & n & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 1 & 0 \\

\end{array}

\right)$ for $\displaystyle n\in\mathbb{N}$.

However, I could not figure out how to compute it by using the eigenvalues.

I would be glad if you show me the solution in this way.

Thanks.

**bkarpuz**