Logarithm of A Matrix - Example: Logarithm of Rotations in The Plane

Example: Logarithm of Rotations in The Plane

The rotations in the plane give a simple example. A rotation of angle α around the origin is represented by the 2×2-matrix

$A = \begin{pmatrix} \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \\ \end{pmatrix}.$

For any integer n, the matrix

$B_n=(\alpha+2\pi n) \begin{pmatrix} 0 & -1 \\ 1 & 0\\ \end{pmatrix},$

is a logarithm of A. Thus, the matrix A has infinitely many logarithms. This corresponds to the fact that the rotation angle is only determined up to multiples of 2π.

In the language of Lie theory, the rotation matrices A are elements of the Lie group SO(2). The corresponding logarithms B are elements of the Lie algebra so(2), which consists of all skew-symmetric matrices. The matrix

$\begin{pmatrix} 0 & 1 \\ -1 & 0\\ \end{pmatrix}$

is a generator of the Lie algebra so(2).

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