Question #146337
Write properties of pauli matrices.
1
Expert's answer
2020-11-27T07:19:36-0500

First let's write what are Pauli matrices :

σ1=(0110),σ2=(0ii0),σ3=(1001)\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][A,B] I note the commutator of AA and BB :

[σ1,σ2]=σ3;[σ2,σ3]=σ1;[σ3,σ1]=σ2[\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

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