FCMatrixProduct[mat1, mat2, ...] can be used to obtain products of matrices with entries containing noncommutative symbols. Using the usual Dot on such matrices would otherwise destroy the original ordering.
The resulting expression can be then further simplified using DotSimplify.
Overview, DataType, DeclareNonCommutative, UnDeclareNonCommutative.
Consider two generic 2 \times 2-matrices containing noncommutative heads
DeclareNonCommutative[opA, opB, opC, opD]mat[1] = {{opA[1], opB[1]}, {opC[1], opD[1]}}
mat[2] = {{opA[2], opB[2]}, {opC[2], opD[2]}}\left( \begin{array}{cc} \;\text{opA}(1) & \;\text{opB}(1) \\ \;\text{opC}(1) & \;\text{opD}(1) \\ \end{array} \right)
\left( \begin{array}{cc} \;\text{opA}(2) & \;\text{opB}(2) \\ \;\text{opC}(2) & \;\text{opD}(2) \\ \end{array} \right)
Using the usual Dot product the elements of the resulting matrix are now multiplied with each other commutatively. Hence, the result is incorrect.
mat[1] . mat[2]\left( \begin{array}{cc} \;\text{opA}(1) \;\text{opA}(2)+\text{opB}(1) \;\text{opC}(2) & \;\text{opA}(1) \;\text{opB}(2)+\text{opB}(1) \;\text{opD}(2) \\ \;\text{opA}(2) \;\text{opC}(1)+\text{opC}(2) \;\text{opD}(1) & \;\text{opB}(2) \;\text{opC}(1)+\text{opD}(1) \;\text{opD}(2) \\ \end{array} \right)
With FCMatrixProduct the proper ordering is preserved
FCMatrixProduct[mat[1], mat[2]]\left( \begin{array}{cc} \;\text{opA}(1).\text{opA}(2)+\text{opB}(1).\text{opC}(2) & \;\text{opA}(1).\text{opB}(2)+\text{opB}(1).\text{opD}(2) \\ \;\text{opC}(1).\text{opA}(2)+\text{opD}(1).\text{opC}(2) & \;\text{opC}(1).\text{opB}(2)+\text{opD}(1).\text{opD}(2) \\ \end{array} \right)
We can also multiply more than two matrices at once
mat[3] = {{opA[3], opB[3]}, {opC[3], opD[3]}}\left( \begin{array}{cc} \;\text{opA}(3) & \;\text{opB}(3) \\ \;\text{opC}(3) & \;\text{opD}(3) \\ \end{array} \right)
out = FCMatrixProduct[mat[1], mat[2], mat[3]]\left( \begin{array}{cc} \;\text{opB}(1).(\text{opC}(2).\text{opA}(3)+\text{opD}(2).\text{opC}(3))+\text{opA}(1).(\text{opA}(2).\text{opA}(3)+\text{opB}(2).\text{opC}(3)) & \;\text{opA}(1).(\text{opA}(2).\text{opB}(3)+\text{opB}(2).\text{opD}(3))+\text{opB}(1).(\text{opC}(2).\text{opB}(3)+\text{opD}(2).\text{opD}(3)) \\ \;\text{opC}(1).(\text{opA}(2).\text{opA}(3)+\text{opB}(2).\text{opC}(3))+\text{opD}(1).(\text{opC}(2).\text{opA}(3)+\text{opD}(2).\text{opC}(3)) & \;\text{opC}(1).(\text{opA}(2).\text{opB}(3)+\text{opB}(2).\text{opD}(3))+\text{opD}(1).(\text{opC}(2).\text{opB}(3)+\text{opD}(2).\text{opD}(3)) \\ \end{array} \right)
Now use DotSimplify to expand noncommutative products
DotSimplify[out]\left( \begin{array}{cc} \;\text{opA}(1).\text{opB}(2).\text{opC}(3)+\text{opB}(1).\text{opC}(2).\text{opA}(3)+\text{opA}(1).\text{opA}(2).\text{opA}(3)+\text{opB}(1).\text{opD}(2).\text{opC}(3) & \;\text{opA}(1).\text{opB}(2).\text{opD}(3)+\text{opA}(1).\text{opA}(2).\text{opB}(3)+\text{opB}(1).\text{opC}(2).\text{opB}(3)+\text{opB}(1).\text{opD}(2).\text{opD}(3) \\ \;\text{opD}(1).\text{opC}(2).\text{opA}(3)+\text{opC}(1).\text{opA}(2).\text{opA}(3)+\text{opC}(1).\text{opB}(2).\text{opC}(3)+\text{opD}(1).\text{opD}(2).\text{opC}(3) & \;\text{opC}(1).\text{opA}(2).\text{opB}(3)+\text{opC}(1).\text{opB}(2).\text{opD}(3)+\text{opD}(1).\text{opC}(2).\text{opB}(3)+\text{opD}(1).\text{opD}(2).\text{opD}(3) \\ \end{array} \right)
Let us define Dirac matrices in the Dirac basis in terms of Pauli matrices
gamma[0] = {{1, 0}, {0, -1}};
gamma[i_] := {{0, CSI[i]}, {-CSI[i], 0}};and express \gamma^i \gamma^j \gamma^i as a 2 \times 2-matrix
FCMatrixProduct[gamma[i], gamma[j], gamma[i]]
DotSimplify[%]\left( \begin{array}{cc} 0.\left(\overline{\sigma }^j.\left(-\overline{\sigma }^i\right)+0.0\right)+\overline{\sigma }^i.\left(0.\left(-\overline{\sigma }^i\right)+\left(-\overline{\sigma }^j\right).0\right) & 0.\left(0.\overline{\sigma }^i+\overline{\sigma }^j.0\right)+\overline{\sigma }^i.\left(\left(-\overline{\sigma }^j\right).\overline{\sigma }^i+0.0\right) \\ 0.\left(0.\left(-\overline{\sigma }^i\right)+\left(-\overline{\sigma }^j\right).0\right)+\left(-\overline{\sigma }^i\right).\left(\overline{\sigma }^j.\left(-\overline{\sigma }^i\right)+0.0\right) & 0.\left(\left(-\overline{\sigma }^j\right).\overline{\sigma }^i+0.0\right)+\left(-\overline{\sigma }^i\right).\left(0.\overline{\sigma }^i+\overline{\sigma }^j.0\right) \\ \end{array} \right)
\left( \begin{array}{cc} 0 & -\overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^i \\ \overline{\sigma }^i.\overline{\sigma }^j.\overline{\sigma }^i & 0 \\ \end{array} \right)