The Equivalent Standard Forms of a Class of Tropical Matrices and Centralizer Groups
In this paper, the equivalent standard forms of tropical idempotent strongly definite matrices are introduced. In particular, the observation of the equivalent standard forms of tropical idempotent normal matrices is given. An equivalence relation <inline-formula><math xmlns="http://ww...
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2025-01-01
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| author | Yanliang Cheng |
| author_facet | Yanliang Cheng |
| author_sort | Yanliang Cheng |
| collection | DOAJ |
| description | In this paper, the equivalent standard forms of tropical idempotent strongly definite matrices are introduced. In particular, the observation of the equivalent standard forms of tropical idempotent normal matrices is given. An equivalence relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> on the set of all tropical idempotent normal matrices, which is relevant to their centralizer groups, is introduced and studied. It is proved that every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-class contains at least one strongly regular tropical idempotent normal matrix. Furthermore, a structural description of the centralizer groups of partial strongly regular tropical idempotent normal matrices is given. |
| format | Article |
| id | doaj-art-a1b83fc6ce644cccb27af7fc79005c63 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-a1b83fc6ce644cccb27af7fc79005c632025-08-20T02:48:02ZengMDPI AGMathematics2227-73902025-01-0113339910.3390/math13030399The Equivalent Standard Forms of a Class of Tropical Matrices and Centralizer GroupsYanliang Cheng0School of Mathematics, Northwest University, Xi’an 710127, ChinaIn this paper, the equivalent standard forms of tropical idempotent strongly definite matrices are introduced. In particular, the observation of the equivalent standard forms of tropical idempotent normal matrices is given. An equivalence relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> on the set of all tropical idempotent normal matrices, which is relevant to their centralizer groups, is introduced and studied. It is proved that every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-class contains at least one strongly regular tropical idempotent normal matrix. Furthermore, a structural description of the centralizer groups of partial strongly regular tropical idempotent normal matrices is given.https://www.mdpi.com/2227-7390/13/3/399tropical semiringtropical matrixidempotent strongly definite matrixidempotent normal matrixcentralizer group |
| spellingShingle | Yanliang Cheng The Equivalent Standard Forms of a Class of Tropical Matrices and Centralizer Groups Mathematics tropical semiring tropical matrix idempotent strongly definite matrix idempotent normal matrix centralizer group |
| title | The Equivalent Standard Forms of a Class of Tropical Matrices and Centralizer Groups |
| title_full | The Equivalent Standard Forms of a Class of Tropical Matrices and Centralizer Groups |
| title_fullStr | The Equivalent Standard Forms of a Class of Tropical Matrices and Centralizer Groups |
| title_full_unstemmed | The Equivalent Standard Forms of a Class of Tropical Matrices and Centralizer Groups |
| title_short | The Equivalent Standard Forms of a Class of Tropical Matrices and Centralizer Groups |
| title_sort | equivalent standard forms of a class of tropical matrices and centralizer groups |
| topic | tropical semiring tropical matrix idempotent strongly definite matrix idempotent normal matrix centralizer group |
| url | https://www.mdpi.com/2227-7390/13/3/399 |
| work_keys_str_mv | AT yanliangcheng theequivalentstandardformsofaclassoftropicalmatricesandcentralizergroups AT yanliangcheng equivalentstandardformsofaclassoftropicalmatricesandcentralizergroups |