ORTHOGONAL

Tell me things about O_n the group of Orthogonal Matrices.

Orthogonal matrices can be defined over any ring as the invertible matrices that preserve some symmetric bilinear form on a free module M. The group is then called O(M), the orthogonal group of M. If the ring has an involution (like complex conjugation) one can also require that the matrices of O(M) should preserve sesquilinear forms. Then one can define U(M) as matrices satisfying (M*)M=I and if 2 is invertible in the ring U(M)=O(M). With these definitions O(C^n)=U(C^n)=U(n) for the standard hermitian inner product.