Answer to Question #146337 in Field Theory for Afraid

Question

Write properties of pauli matrices.

Answer 1

First let's write what are Pauli matrices :

$\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_2=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_3= \begin{pmatrix} 1 & 0 \\0 & -1 \end{pmatrix}$

The main properties are :

• All three are Hermitian and unitary matrices.
• All of them have eigenvalues 1 and -1
• By $[A,B]$ I note the commutator of $A$ and $B$ :

$[\sigma_1,\sigma_2]=\sigma_3; [\sigma_2,\sigma_3]=\sigma_1; [\sigma_3,\sigma_1]=\sigma_2$

These are the main properties of Pauli matrices, but there is, obviously, a lot more of them. You can also check Wikipedia : Pauli matrices - Wikipedia