Originally Posted by
barbos
What do you mean by that? Unitary matrix consists of rows which are ortho-normal to each other.
In other words, any ortho-normal basis corresponds to unitary matrix.
That is a constraint on the matrices, not a specification of the matrices in terms of some parameters.
For the SU(2) matrices, one can find them in terms of the unit quaternions:
D = x0*I - i*(x1*s1 + x2*s2 + x3*s3)
where 4-vector x = (x0,x1,x2,x3) has unit length, x.x = 1, and obeys the quaternion multiplication law. The matrices s1, s2, s3 are the Pauli matrices: {{0,1},{1,0}}, {{0,-i},{i,0}}, {{1,0},{0,-1}}. The matrix I is the identity matrix: {{1,0},{0,1}}.
There is a similar expression for the SO(3) matrices, but it is more complicated, and it contains a sort of square of the quaternion vector.
Rotation matrix
Quaternions are often used in computer 3D graphics, because it is relatively easy to smoothly change them -- it is much easier to keep a vector normalized than a matrix orthonormalized.