Centralizers, Clifforders, Polynomial Equivalence and $\omega$-equivalence of Matrices

This article is devoted to the study of the centralizer and the clifforder of a matrix over a field $\mathbb{F}$ of characteristic zero, as well as the quasi-commutative relations between matrices over the complex field $\mathbb{C}$. We introduce several new concepts, including polynomial equivalence, odd polynomial equivalence, $q$-polynomial equivalence, the clifforder of a matrix, balanced matrices, and $\omega$-equivalence. The clifforder of a matrix is defined via an anti-commuting relation. We present a new proof establishing that two matrices $A$ and $B$ share the same centralizer if and only if they are in polynomial equivalence. Moreover, we extend this to a broader generalization. For balanced matrices (including nilpotent matrices), we prove that their clifforders coincide if and only if they are odd polynomial equivalence. In addition, we investigate quasi-commutative relations defined using a primitive $q$-th root of unity $\omega$, provide a new and elementary proof of a classical theorem of H. S. A. Potter, and further explore the properties and connections of $\omega$-equivalence.