How to recover the Logarithm of rotations in the plane

Question

Let

theta is the angle of rotation.

The Logarithm of this matrix is:

The way to prove this is just take the exponential of B:

exp(B); however I have no idea how to do it to obtain A?

I try to use diagonalization: (suppose n = 0)

How to go further? the first term and third term is one problem.

Answer 1

I find an efficient way to solve this. Instead of using diagonalization, the approach is as following:

1. Find inv(sI - A), where s is the Laplacian operator and I is the identity matrix.

2. Apply the fomula for L^-1(inv(sI - A)), we obtain:

   s/(s^2 + 1) = cos(theta)
   -1/(s^2 + 1) = -sin(theta) ...

Done!

I always consider the Diag. when facing matrix exponential; however, Laplas TF is sometimes a better and quick way.

Answer 2

You could use the Taylor series of $\exp$ (hint: the powers of $\left[ \matrix{0 & -1\cr 1 & 0\cr} \right]$ have a simple pattern). Or diagonalize...