Let be the matrix with 1 in the -th entry and 0 elsewhere. Consider the space of complex matrices, , for a fixed d. Define the following matrices
For , .
For , .
Let , the identity matrix.
For , .
For , .
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-Schmidtinner product on . By the dimension count, we see that they span the vector space of complex matrices.
In dimensions 2 and 3, the above construction recovers the Pauli and
Gell-Mann matrices, respectively.