Pauli Matrices - Algebraic Properties

Algebraic Properties

\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = -i\sigma_1 \sigma_2 \sigma_3 = \begin{pmatrix} 1&0\\0&1\end{pmatrix} = I

where I is the identity matrix, i.e. the matrices are involutory.

  • The determinants and traces of the Pauli matrices are:

From above we can deduce that the eigenvalues of each σi are ±1.

  • Together with the identity matrix I (which is sometimes written as σ0), the Pauli matrices form an orthogonal basis, in the sense of Hilbert–Schmidt, for the real Hilbert space of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices.

Read more about this topic:  Pauli Matrices

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