Generalizations of Pauli Matrices - A Non-Hermitian Generalization of Pauli Matrices | Non-Hermitian Generalization Pauli Matrices

A Non-Hermitian Generalization of Pauli Matrices

The Pauli matrices and satisfy the following:

$\sigma _1 ^2 = \sigma _3 ^2 = I, \; \sigma _1 \sigma _3 = - \sigma _3 \sigma _1 = e^{\pi i} \sigma _3 \sigma_1.$

The so-called Walsh-Hadamard conjugation matrix is

$W = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}.$

Like the Pauli matrices, W is both Hermitian and unitary. and W satisfy the relation

The goal now is to extend above to higher dimensions, a problem solved by J. J. Sylvester (1882).