A simplification regarding matrices Let $$A = \left( \begin{array}{cc}
0 & -1   \\
1 & 0   \\
   \end{array} \right )$$
What can I say about the matrix $A^k$ for arbitrary natural $k$. Is there some way to express $A^k$ in terms of $A$. I have observed that $A^4 = I$.
EDIT: I want a better form of $A^k$ in order to find a closed form expression in terms of elementary functions for $\exp(A)$

We can diagonalize $A$ by calculating the Jordan Normal Form, using it's eigenvalues, $\lambda_{1, 2} = \pm i$, where $i$ is the imaginary unit for this matrix as:
$$A = \left( \begin{array}{cc}
0 & -1   \\
1 & 0   \\
   \end{array} \right ) = S \cdot J \cdot S^{-1} = \left( \begin{array}{cc}
-i & \ i   
\\ 1  & 1 \\
\end{array} \right ) \left( \begin{array}{cc}
-i & \ 0   
\\ 0  & i \\
\end{array} \right ) \left( \begin{array}{cc}
\frac{i}{2} & \ \frac{1}{2}   
\\ \frac{-i}{2}  & \frac{1}{2} \\
\end{array} \right )$$
We can now calculate the exponential of $A$ using 
$$\exp(A)=S \cdot \exp(J) \cdot S^{-1}$$
Regards
