# Encoding quaternions as matrices?

• Mar 20th 2011, 06:08 AM
EuropeanSon
Encoding quaternions as matrices?
I've a problem from my Group theory module that's been puzzling me, which is asking me to rotate a vector x (1,0,0) around a vector Ov, where v is (1/sqrt(3), -1/sqrt(3), 1/sqrt(3)), by 60 degrees by representing x as a matrix Z (encoding (0,1,0,0) as a matrix) and computing
UZU*,
where U is the unitary matrix encoding the quaternion
cos30 + vsin30.

Now, I know how to rotate it using quaternion conjugates, and how to do it using the cross product method, but this terminology is alien to me. What is a Unitary matrix and how is it calculated? Our lecture notes don't cover it, nor do they mention "encoding quaternions as matrices". I have no idea where to start. Any pointers?

Thanks for any help that is forthcoming.
• Mar 21st 2011, 03:09 AM
Ackbeet
I am by no means an expert here, but this link might prove useful. Compare the quaternion method at the end with the matrix multiplication method at the beginning. I can't help you any more than that, though.
• Mar 21st 2011, 11:06 AM
EuropeanSon
Thanks. I got it done in the end, the solution was to represent each dimension of the quaternion as a multiple of specific 2x2 matrices, namely
1=I
i=
[i 0]
[0 -i]
j=
[0 1]
[-1 0]
k=
[0 i]
[i 0]

And then add them together to get the appropriate matrix.
• Mar 21st 2011, 11:17 AM
Ackbeet
Ah, the Pauli spin matrices strike again. Glad you got it.