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
For any integer n, the matrix
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
is a generator of the Lie algebra so(2).
Read more about this topic: Logarithm Of A Matrix
Famous quotes containing the word plane:
“Even though I had let them choose their own socks since babyhood, I was only beginning to learn to trust their adult judgment.. . . I had a sensation very much like the moment in an airplane when you realize that even if you stop holding the plane up by gripping the arms of your seat until your knuckles show white, the plane will stay up by itself. . . . To detach myself from my children . . . I had to achieve a condition which might be called loving objectivity.”
—Anonymous Parent of Adult Children. Ourselves and Our Children, by Boston Womens Health Book Collective, ch. 5 (1978)