Gell-Mann matrices

The Gell-Mann matrices are denoted by ${\lambda }_1\dots {\lambda }_8$. They form a family of traceless Hermitian $(3\times 3)$- matrices, orthonormalized as follows: $\mathrm{Tr}({\lambda }_j{\lambda }_k)=2{\delta }_{jk}$. When multiplied by the complex unit they form a basis in the Lie algebra $\mathfrak{s}\mathfrak{u}(3)$, in analogy with the Pauli matrices and the Lie algebra $\mathfrak{s}\mathfrak{u}(2)$. Their explicit form is [a1]:

$${\lambda }_1=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),{\lambda }_2=\left(\begin{array}{ccc}0& -i& 0\\ i& 0& 0\\ 0& 0& 0\end{array}\right),$$

$${\lambda }_3=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right),{\lambda }_4=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right),$$

$${\lambda }_5=\left(\begin{array}{ccc}0& 0& -i\\ 0& 0& 0\\ i& 0& 0\end{array}\right),{\lambda }_6=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right),$$

$${\lambda }_7=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& -i\\ 0& i& 0\end{array}\right),{\lambda }_8=\frac{1}{\sqrt{3}}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -2\end{array}\right).$$

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