Understanding the Normalization of Pauli Matrix in Quantum Mechanics

[Lizwi]

Why is norm of (pauli matrix)/sqrt(2)=1

 

[HallsofIvy]

Which "Pauli matrix" are you talking about? My first thought was the "Pauli matrices" used in quantum mechanics:
\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}
but they all have determinant -1.

 

[Fredrik]

The set of Pauli matrices is a basis for the (real) vector space of complex traceless self-adjoint 2×2 matrices. If we define the inner product on that space by $\langle A,B\rangle=\operatorname{Tr}(A^*B)$, where * denotes conjugate transpose, then I think the matrices $E_i=\sigma_i/\sqrt{2}$ form an orthonormal basis of that space. (You should check to make sure that I remember this right).

 

[HallsofIvy]

Ahh! so the key point is that for every Pauli matrix, A, A*A= the 2 by 2 identity matrix that has trace 2.