# How to get Gell-Mann matrices?

How to get Gell-Mann matrices $f_{i}$ (more or less strictly)? What are the requirements for getting them, excluding $||f_{i}|| = 1$, commutational law $[f_{i}, f_{j}] = if_{ijk}f_{k}$ and hermitian nature? Is that enough?

• I show how to do the diagonal ones here: math.stackexchange.com/q/4100876/202425. The rest are similar (they come from the Pauli matrices, padded with zeros). Apr 13, 2021 at 19:01

It may be got by a simple properties of $$SU(N)$$ group matrix representation.
First, by representing the matrix $$\hat {U}$$ of group near identity matrix, $$\hat {U} = \hat {E} + \hat {A}$$, you may (keeping the linearity by $$\hat {A}$$) get properties of $$\hat {A}$$: $$\hat {U}\hat {U}^{+} = \hat {E} \Rightarrow \hat {A}^{+} = -\hat {A}, \quad det \hat {U} = 1 \Rightarrow Tr(\hat {A}) = 0.$$ The first condition leads to the absence of the real part of diagonal components of $$\hat {A}$$ and to representation of non-diagonal components in form of $$A_{ij} = a_{ij} + ib_{ij}, A_{ji} = -a_{ij} + ib_{ij}$$. The second leads to condition of $$\sum_{i} A_{ii} = 0$$, but except this the parametrization of the diagonal components is arbitrary.
So for $$SU(3)$$ representation is not hard to see that corresponding matrice is $$\hat {A} = i\begin{pmatrix} a_{3} + a_{8} & a_{1} - ia_{2} & a_{4} - ia_{5} \\ a_{1} + ia_{2} & a_{8} - a_{3} & a_{6} - ia_{7} \\ a_{4} + ia_{5} & a_{6} + ia_{7} & -2a_{8} \end{pmatrix}.$$ It remains only to expand matrix in sum of $$\sum_{i}a_{i}\hat {R}_{i}$$, where $$\hat {R}_{i}$$ are Gell-Mann matrices.