John Stillwell - Naive Lie Theory.pdf - Index of

52 3 Generalized rotation groups
An example of a transformation that is in O(n), but not in SO(n), isreflection
in the hyperplane orthogonal to the x 1 -axis, (x 1 ,x 2 ,x 3 ,...,x n ) ↦→
(−x 1 ,x 2 ,x 3 ,...,x n ), which has the matrix
⎛
⎞
−1 0 ... 0
0 1 ... 0
⎜
⎟
⎝ .
⎠ ,
0 0 ... 1
obviously of determinant −1. We notice that the determinant of a matrix
A ∈ O(n) is ±1 because (as mentioned in the previous section)
AA T = 1 ⇒ 1 = det(AA T )=det(A)det(A T )=det(A) 2 .
Path-connectedness
The most striking difference between SO(n) and O(n) is a topological one:
SO(n) is path-connected and O(n) is not. That is, if we view n×n matrices
as points of R n2 in the natural way—by interpreting the n 2 matrix entries
a 11 ,a 12 ,...,a 1n ,a 21 ,...,a 2n ,...,a n1 ,...,a nn as the coordinates of a point—
then any two points in SO(n) may be connected by a continuous path in
SO(n), but the same is not true of O(n). Indeed, there is no continuous
path in O(n) from
⎛
⎞ ⎛
⎞
1
−1
1
⎜
⎝
. ..
⎟
⎠ to 1
⎜
⎝
. ..
⎟
⎠
1
1
(where the entries left blank are all zero) because the value of the determinant
cannot jump from 1 to −1 along a continuous path.
The path-connectedness of SO(n) is not quite obvious, but it is interesting
because it reconciles the everyday concept of “rotation” with the
mathematical concept. In mathematics, a rotation of R n is given by specifying
just one configuration, usually the final position of the basis vectors,
in terms of their initial position. This position is expressed by a matrix
A. In everyday speech, a “rotation” is a movement through a continuous
sequence of positions, so it corresponds to a path in SO(n) connecting the
initial matrix 1 to the final matrix A.