Normalization of a spin-like quantity in matrix mechanics

Question

Suppose that there is a quantity in Heisenberg picture as the following:

$A=u_1\Sigma_1 + u_2\Sigma_2 +u_3\Sigma_3$

I am not sure why $u_1,u_2,u_3$ is normalized to be ${u_1}^2 + {u_2}^2 + {u_3}^2 =1$. (The matrices $\Sigma$ are the Pauli matrices.)

Answer 1

This is because you are doing a rotation transformation, which is an SU(2) on the state vector (the vector the matrices act on), but when you do the transformation of the Pauli operators, it is a regular vector rotation of all three, so it preserves the length. This is a special case for 2 state quantum mechanics, any SU(2) transformation preserves the length of the coefficients of the Pauli matrix in the expansion of any 2 by 2 matrix, the Pauli matrices make an operator vector.