Editing Gell-Mann matrices (section)

===Fierz completeness relations===

Since the eight matrices and the identity are a complete trace-orthogonal set spanning all 3×3 matrices, it is straightforward to find two Fierz '''''completeness relations''''', (Li & Cheng, 4.134), analogous to that [[Pauli matrices#Completeness relation 2|satisfied by the Pauli matrices]]. Namely, using the dot to sum over the eight matrices and using Greek indices for their row/column indices, the following identities hold,
:<math>\delta^\alpha _\beta  \delta^\gamma  _\delta   = \frac{1}{3} \delta^\alpha_\delta  \delta^\gamma _\beta  +\frac{1}{2} \lambda^\alpha _\delta \cdot \lambda^\gamma _\beta </math>
and 
:<math>\lambda^\alpha _\beta \cdot  \lambda^\gamma  _\delta   = \frac{16}{9} \delta^\alpha_\delta  \delta^\gamma _\beta  -\frac{1}{3} \lambda^\alpha _\delta \cdot \lambda^\gamma _\beta ~.</math>
One may prefer the recast version, resulting from a linear combination of the above,
:<math>\lambda^\alpha _\beta \cdot  \lambda^\gamma  _\delta   = 2 \delta^\alpha_\delta  \delta^\gamma _\beta  -\frac{2}{3}  \delta^\alpha_\beta  \delta^\gamma _\delta    ~.</math>