User:Marozols/Sandbox

Construction
Let $$\; E_{jk}$$ be the matrix with 1 in the $$jk$$-th entry and 0 elsewhere. Consider the space of $$d \times d$$ complex matrices, $$\mathbb{C}^{d \times d}$$, for a fixed d. Define the following matrices


 * For $$ k < j$$, $$f_{k,j} ^d = E_{kj} + E_{jk}$$.


 * For $$ k > j$$, $$f_{k,j} ^d = - i ( E_{jk} - E_{kj} )$$.


 * Let $$h_1 ^d = I_d $$, the identity matrix.


 * For $$1 < k < d$$, $$h_k ^d = h ^{d-1} _k \oplus 0$$.


 * For $$k = d$$, $$h_d ^d = \sqrt{\frac{2}{d(d-1)}} (h_1 ^{d-1} \oplus (1-d))$$.

The collection of matrices defined above are called the generalized Gell-Mann matrices, in dimension d.

Properties
The generalized Gell-Mann matrices are Hermitian and traceless by construction, just like the Pauli matrices. One can also check that they are orthogonal in the Hilbert-Schmidt inner product on $$\mathbb{C}^{d \times d}$$. By the dimension count, we see that they span the vector space of $$d \times d$$ complex matrices.

In dimensions 2 and 3, the above construction recovers the Pauli and Gell-Mann matrices, respectively.