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 ?
 A: If a linear transformation $T:\Bbb R^3\to\Bbb R^3$ is a non-trivial rotation, then the set $\{x\in\Bbb R^3:T(x)=x\}$ will be the axis of rotation, since non-trivial rotation about an axis moves every point except the points on the axis. Also, if the determinant of $T$ isn't $1,$ then it isn't a rotation (why?), though you've already seen that $\det(A)=1$ in this case.
Here, we're working with the transformation $T(x)=Ax$, so the set $\{x\in\Bbb R^3:T(x)=x\}$ is just the eigenspace of $A$ corresponding to the eigenvalue $1$. If $A$ didn't have $1$ as an eigenvalue, we'd know it wasn't a rotation at all (in this case, it does have $1$ as an eigenvalue). If the eigenspace's dimension were greater than $1$, then either it'd be a reflection matrix (if dimension $2$), or the identity matrix (if dimension $3$). The latter is clearly not the case, so it's either a rotation matrix or a reflection matrix. However, if it were a reflection matrix, its determinant would be $-1,$ instead (why?), and so it is a rotation matrix.

Side Note: Given any two non-zero vectors $x,y$ in $\Bbb R^3$ with the angle from $x$ to $y$ being $\theta$, we may assume without loss of generality that $\theta$ is no less than $0$ radians and no more than $\pi$ radians. (Why?) Then we have the following formula (where $\cdot$ is the dot product): $$x\cdot y=\lVert x\rVert\lVert y\rVert\cos\theta\tag{1}$$
To see where $(1)$ comes from, see here.

In general, let's suppose we've been given some matrix $A$ corresponding to a rotation in $\Bbb R^3$, and that we want to find its angle of rotation. First, find a basis $\{w\}$ for the axis of rotation (found as above), let $x$ be any non-zero unit vector orthogonal (perpendicular) to $w$, let $y=Ax$. Then both $x$ and $y=Ax$ will be unit vectors. (Do you see why $y$ is a unit vector?), so formula $(1)$ yields the following alternate formula for our particular $x,y$: $$x\cdot y=\cos\theta\tag{$1'$}$$ Here, $\theta$ is the angle of rotation of $A$. (Do you see why?) From there, we can determine $\theta$. (Do you see why and how?)
Alternatively, start with $w$ (as above), normalize it to $\hat w$, and then determine an orthonormal basis $B=\{\hat w,v_2,v_3\}$ for $\Bbb R^3$ with the Gram-Schmidt process. Then $$(\hat w\: v_2\: v_3)^TA(\hat w\: v_2\: v_3)=\left(\begin{array}{ccc}1 & 0 & 0\\0 & \cos\theta & -\sin\theta\\0 & \sin\theta & \cos\theta\end{array}\right),$$ which gives us another way to find $\theta$.
A: Wolfram Alpha tells me there is an eigenspace with eigenvalue $-1$ generated by $(-2, 0, 1)$ and $(-2, 1, 0)$ and an eigenspace with eigenvalue $1$ generated by $(1, 2, 2)$. (You could do this by hand). The eigenspaces are orthogonal, so this is a rotation by 180 degrees about the axis $(1, 2, 2)$. 
The problem is a little atypical, since if you rotate by any angle that's not an integer multiple of $\pi$ you will have complex eigenvalues.
A: A rotation matrix has unit determinant.  Such a matrix that has all non-zero entries may be decomposed into three rotation matrices, each representing a rotation about an orthogonal coordinate axis.
A: 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 $\pi$.
A: The simplest way to find the rotation angle is to take the trace of the matrix, the sum of the diagonal elements. By Cameron Buie's answer this equals $1 + 2\cos(\theta)$ where $\theta$ is the angle of rotation. $\theta$ can then be determined up to sign which will depend on the orientation of the axis of rotation chosen.
For non-symmetric matrices, the axis of rotation can be obtained from the skew-symmetric part of the rotation matrix, $S = .5(R-R^\mathrm{T})$;
Then if $S=(a_{ij})$, the rotation axis with magnitude $\sin\theta$ is $(a_{21},a_{02},a_{10})$.
A: The document: orthogonal matrices
is an excellence reference for this problem.
The steps are as follows.


*

*Show that the determinant is 1. Matrices with determinant -1 are
reflections no rotations.

*Find the eigenvalues.   The three eigenvalues of the matrix are $1,
    \text{e}^{-i \theta}$,  where $\theta$ is the angle of rotation.

*find the eigenvector for the eigenvalue 1. This is the axis of
rotation.
Solution:
 That the determinant is 1 can be directly checked.
The eigenvalues are $\lambda=1, \pm 1$. 
  So the angle of rotation is  $0$.
The eigenvector for the eigenvalue $1$ is
  $(1/2,1,1)$. This is the axis of rotation.
