The generalized GellMann matrices are the matrices generating the Lie algebra associated to the special unitary group , . As their name suggests, these matrices are intended to generalize both the standard GellMann matrices, which generate the Lie algebra associated to , as well as the Pauli matrices which generate the Lie algebra associated to .
The algorithm for constructing the generalized GellMann matrices is as follows. Throughout, let denote the matrix with a 1 in the th entry and 0 elsewhere. This allows one to define three collections of matrices. The first collection is symmetric:
(1)

for . The second collection is antisymmetric:
(2)

for . The third collection is diagonalDiagonal Matrix:
(3)

for .
This gives a total of
(4)

generalized GellMann matrices, matching exactly the realdimension of the Lie algebra associated to .
Note that the construction can be rephrased using braket notation (Bertlmann and Krammer 2008) as well. In addition, one can easily verify that the cases for and yield the Pauli and GellMann matrices, respectively.