An

*nxn* matrix over a

field (for example the

real numbers
or

complex numbers) is called

orthogonal if

*A.A*^{t}=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 **R**^{n}
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.