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.