A 2x2 matrix whose elements are complex numbers. I don't think there are many practical advantages in using Paul matrices for classical 3D rotations. Each rotation requires 2x2 complex numbers which requires 8 scalar numbers, nearly as many as using 3x3 scalar matrix for rotations which most people would consider much simpler. I think Pauli matrices are interesting though, because instead of having to learn the rules for multiplying the operators i, j and k these multiplication rules come automatically provided you know how to multiply 2x2 matrices. If you want to calculate the spin of fundamental particles in many dimensions then Pauli matrices may be the only way to do it.
The matrix is generated by multiplying the three imaginary values by the following Pauli matrices:
The real value is multiplied by the identity matrix.
The Pauli matrices in this form are not the exact equivalent of quaternions this is because, if we square them, we get +1 and not -1. In other words the identity matrix:
| t12 = |
|
* |
|
= |
|
| t22 = |
|
* |
|
= |
|
| t32 = |
|
* |
|
= |
|
So these dimensions are given by √+1instead of √-1which quaternions use. In this form Pauli matrices have different properties, they don't form a normed division algebra. However we can convert Pauli matrices to have exactly the same properties to quaternions by multiplying by i. Then each of the generators are √-1 as shown here:
| t12 = |
|
* |
|
= |
|
| t22 = |
|
* |
|
= |
|
| t32 = |
|
* |
|
= |
|
Pauli matrices were developed for physics (quantum mechanics) so that may be why they are formulated with this i factor difference from quaternions.
In quantum mechanics a + i b + j c + k d and -a - i b - j c - k d
represent different spins for particles, so a particle has to rotate through
720 degrees instead of 360 degrees to get back where it started. Both Pauli matrices and quaternions have this property, see Spinors.
