# Thread: Fun in the SU(N)

1. Originally Posted by lpetrich  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.
Still don't understand your complain about not having an expression when you yourself gave said expression, not to mention me giving it earlier.  Reply With Quote

2. barbos, what you describe are constraints. I will be more explicit:

Orthogonality:
M.MT = MT.M = I
MT = transpose of M

Unitarity:
M.M+ = M+.M = I
M+ = Hermitian conjugate of M = transpose complex conjugate of M

Another kind: symplecticity:
M.J.MT = J
where
J = {{0, I}, {-I, 0}}
where J has size 2n * 2n and the I's and 0's in it have size n * n.

Special matrices: determinant of them = 1  Reply With Quote

3. Here are some group and algebra interrelationships.

U(n) ~ SU(n) * U(1)

Group O(n) has subgroup SO(n) with quotient group Z2 {1, -1}

O(n) ~ SO(n) * Z2 only for odd n

• SO(1) ~ empty algebra, identity group
• SO(2) ~ U(1)
• SO(3) ~ SU(2) ~ Sp(2)
• SO(4) ~ SU(2) * SU(2)
• SO(5) ~ Sp(4)
• SO(6) ~ SU(4)

Sp = symplectic group / algebra  Reply With Quote

4. Originally Posted by lpetrich barbos, what you describe are constraints. I will be more explicit:
Nope, I wrote the expression A=exp(i*H) where H is any hermitian operator which itself is a linear combination of what is called generator operators.  Reply With Quote

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