Axis and angle of rotation from 3x3 rotation matrix and properties of rotation matrix

Question

A 3x3 rotation matrix is given. We know the properties of rotation matrix:

• The columns of rotation matrix are unit vectors perpendicular to each other.
• The column vectors indicate where the unit vectors along the principal axes are transformed.
• The rows of rotation matrix are unit vectors perpendicular to each other.
• The row vectors indicate the vectors that are transformed into the unit vectors along the principal axes.
• The inverse of rotation matrix is its transpose.

Suppose, we don't know anything about the Rodrigues formula, so we can't use Trace(R) = 1+2cos$\theta$. Is it possible to find the axis and angle of rotation from the given rotation matrix and the properties of rotation matrix? I need an answer with proper explanation.

Answer 1

If $A$ is your rotation matrix, then the axis can be found by solving the vector equation $Ax=x$. Take a unit vector $u$ along the axis, another unit vector $v$ perpendicular to $u$ (say by Gram-Schmidt) and $w=u\times v$ (vector product). Then $Av=(\cos\theta )v+(\sin\theta)w$ for some $\theta$ (that's the angle of rotation).

The right way to find $\theta$ is of course to use $\text{trace}(A)=1+2\cos\theta$.