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

This paper is devoted to the study of the centralizer and the clifforder of matrices over a field $\mathbb{F}$ of characteristic zero, together with the quasi-commutative relations between them.

Several new notions are introduced, including polynomial equivalence, odd polynomial equivalence, $q$-polynomial equivalence, the clifforder of a matrix, balanced matrices, and $\omega$-equivalence.

We also define the $k$-th annihilator of a matrix and the $k$-fold composition of the adjoint operator.

Using these concepts, we extend the classical double centralizer theorem to a broader framework, showing that the classical case arises as a special instance.

For balanced (including nilpotent) matrices, we prove that their clifforders coincide if and only if they are odd polynomial equivalence.

Moreover, we provide another proof of a theorem of H.

S.

A.

Potter by using quasi-commutative relations defined by a primitive $q$-th root of unity $\omega$, as well as another proof of several further known results on $\omega$-equivalence.