Centralizers, Clifforders, Polynomial Equivalence and $ω$-equivalence of Matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912656742416384 |
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| author | Zhang, Hechun Zhu, Chengyi |
| author_facet | Zhang, Hechun Zhu, Chengyi |
| contents | 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 $ω$-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 $ω$, provide a new and elementary proof of a classical theorem of H. S. A. Potter, and further explore the properties and connections of $ω$-equivalence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15932 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Centralizers, Clifforders, Polynomial Equivalence and $ω$-equivalence of Matrices Zhang, Hechun Zhu, Chengyi General Mathematics 15A18, 15A27 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 $ω$-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 $ω$, provide a new and elementary proof of a classical theorem of H. S. A. Potter, and further explore the properties and connections of $ω$-equivalence. |
| title | Centralizers, Clifforders, Polynomial Equivalence and $ω$-equivalence of Matrices |
| topic | General Mathematics 15A18, 15A27 |
| url | https://arxiv.org/abs/2510.15932 |