An
nxn matrix over a
field (for example the
real numbers
or
complex numbers) is called
orthogonal if
A.At=I. That is,
A is invertible and its
inverse is its
transpose.
The orthogonal group O(n,k)
is the group of nxn orthogonal matrices over k
In the case of the real numbers k=R, this
group is the group of linear isometries of Rn
and so has an obvious importance in geometry. The case of
n=2 and and n=3 are particularly significant for physical
reasons.
O(n,k) has a normal subgroup (of index two) called the special
orthogonal
group, SO(n,k). This consists of those othogonal matrices with
determinant 1.
From now on we'll consider the case k=R and just
write O(n) and SO(n). These are examples of Lie groups.
O(2) consists of two kinds of matrices
-- -- -- --
S(a)= | cos a sin a | R(a)= | cos a -sin a |
| sin a -cos a | | sin a cos a |
-- -- -- --
The first of these is a
reflection in a line through the origin
which makes an angle of
a/2 with the x-axis. The second
is rotation anticlockwise through the angle
a about
the origin.
SO(2) consists of rotations.
SO(3) again consists of rotations,
this time about some line through the origin
(an axis). This is a result of Gauss.