Find the rotation axis and angle of a matrix $$A=\frac{1}{9}
\begin{pmatrix}
-7 & 4 & 4\\ 
4 & -1 & 8\\ 
4 & 8 & -1
\end{pmatrix}$$
How do I prove that A is a rotation ? How do I find the rotation axis and the rotation angle ?

You have $A^T A = I$. Hence $A$ is a rotation. Since $\det A = 1$, it is proper.
By inspection, $A \begin{bmatrix} 1 \\ 2 \\ 2\end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\2\end{bmatrix}$, which gives the axis of rotation.
Inspection also shows that $\begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 4 \\ -5 \end{bmatrix}$ are orthogonal eigenvectors corresponding to the (repeated) eigenvalue $-1$. Hence we see that the rotation angle is $\pi$.
Explicitly, if we let $R = \begin{bmatrix} 1 & 2 & 2 \\ 2 & -1 & 4 \\ 2 & 0 & -5 \end{bmatrix}$, then  $R^{-1} = \frac{1}{405} \begin{bmatrix} 45 & 90 & 90 \\  162 & -81 & 0 \\ 18 & 36 & -45\end{bmatrix}$, and $R^{-1} A R = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$, from which we see that the rotation angle is $\pi$.
