Let $A$ be the following $3 \times 3$ matrix:
$$ A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix} $$
I'm supposed to calculate $A^n$, where $n \in \Bbb R$, $\exp(tA)$ and $\sin(\pi A)$. Obviously $A$ is not diagonalizable. Since we haven't had anything about Jordan decomposition in the lecture, I'm not sure how to solve this.
The eigenvalues $\lambda_1 = -1 , \lambda_{2,3} = 1$ can be read off. I tried to expand the two eigenvectors into a orthonormal basis, i.e.:
$$ \mathbf{x}_{\lambda_1} = \begin{pmatrix} 0 \\ 0 \\ 1 \\ \end{pmatrix} \qquad \mathbf{x}_{\lambda_2} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix} \qquad \mathbf{x}_3 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ \end{pmatrix} $$
But I'm rather unsure how to continue. I suspect that
$$ A^n = \begin{pmatrix} 1 & n & 0 \\ 0 & 1 & 0 \\ 0 & 0 & (-1)^n \\ \end{pmatrix} \qquad \text{for} \qquad n \in \Bbb N_0, $$
But how to expand this to $n \in \Bbb R$? In general, how can I solve such a problem of matrix functions, if I've not heard anything about Jordan decomposition?
EDIT: Thanks for your help. I could show that the above mentioned matrix for $A^n$ is correct even for $n \in \Bbb Z$. The two other functions are straightforward then. If someone has an idea or hint about $A^n$ for $n \notin \Bbb Z$, i would appreciate it.