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We construct two sets of representations of the Dirac equation. Theytransform themselves from one to another in multiplying by the tensorcommutation matrix (TCM) $2\otimes 2$. The Gauss matrix can lead us to theCholesky decomposition. The TCMs can be used in order to obtain differenttransforms of some matrix equations to linear matrix equations of the form$AX=B$. A TCM $n\otimes n$ is expressed in terms of the $n\otimes n$-Gell-Mannmatrices. In order to generalize this expression to which of TCMs $n\otimes p$,we introduced what we call rectangle Gell-Mann matrices. The electric chargeoperator (ECO) for eight leptons and quarks of the Standard Model (SM) of asame generation proposed by Zenczykowski is expressed in terms of the TCM$2\otimes 2$. An ECO including all the fermions of the SM is constructed interms of TCM $4\otimes 4$. Then the eigenvalues of a TCM take sense.

Matrices de commutation tensorielle: de l'\'equation de Dirac vers une application en physique des particules