How to tell what rotation operation a matrix performs?

Question

For example, a matrix $ M= \begin{bmatrix} cos^2(a) & -sin(a) & -cos(a)sin(a)\\ cos(a)sin(a) & cos(a) & -sin^2(a) \\ sin(a) & 0 & cos(a) \end{bmatrix}$. What is the axis and angle of this rotation?

Answer 1

Take a look at this link. Section "Formalism Alternatives" subsection "Rotation Matrix" shows how to get the rotation angle and axis from the matrix.

Answer 2

The axis will be in the direction of the eigenvector of $M$ for the eigenvalue $1$.

The angle is recoverable from the equation $Tr(M)=1+2\cos(\theta)$. When solving for $\theta$, you'll have to be careful to get the sign of rotation to match the eigenvector you've chosen.

Answer 3

Your axis is given by the eigen vector $v$ such as $Mv=v$ (eigen value : $1$)

Let $u$ be a vector perpendicular to $v$ then $cos(\theta)=\frac{(Mu,u)}{||u||^2}$