While looking into the camera rotation dataset, I had found a $3\times3$ rotation matrix $R$ which has very slight change from original : $$R=\begin{bmatrix} 0.99995284&-0.01584106&-0.01266612\\ -0.0045012&0.99991074&-0.01571224\\ 0.01275538&0.01563972&0.99979431\end{bmatrix}$$
I'd like to calculate the distance this one from the $3\times3$ identity matrix but had never learned a unary operation which maps $\Bbb R^{m\times n} \to \Bbb R$.
Which way could I adpot to calculate one?
The rotation matrices form a Lie group, which is a manifold in the set of all $3\times 3$ matrices. Ideally, you need a metric which gives you the geodesic distance between two matrices.
The tangent space of the $3\times 3$ rotation matrices is the set of $3\times 3$ skew-symmetric matrices, which can be obtained using the exponential map. This allows us to define the geodesic metric using the tangent space, $$ \rho(R_1,R_2) = \frac{1}{\sqrt{2}}\| \log(R_1^{-1}R_2)\|_F $$
Note that the $\log$ function is the matrix logarithm.