| 广义二次矩阵的广义多项式秩不变性 |
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| 引用本文： | 陈梅香,杨忠鹏,林志兴,冯晓霞.广义二次矩阵的广义多项式秩不变性[J].吉林大学学报(理学版),2022,60(2):253-260. |
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| 作者姓名： | 陈梅香 杨忠鹏 林志兴 冯晓霞 |
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| 作者单位： | 1. 莆田学院 应用数学福建省高校重点实验室, 福建 莆田 351100；
2. 闽南师范大学 数学与统计学院, 福建 漳州 363000 |
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| 基金项目： | 福建省自然科学基金项目;国家自然科学基金 |
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| 摘 要： | 首先, 利用表示为(A-dP)(A-eP)=0的广义二次矩阵A与幂等矩阵P的关系, 讨论A的广义多项式fP(A)的基本性质, 并证明广义多项式运算的秩不变性. 结果表明, 广义多项式的秩不仅与组合系数的选择无关, 而且在大多数情形下与多项式的选择也无关. 其次, 作为应用, 概括并推广已有幂等矩阵、对合矩阵、二次矩阵、 广义二次矩阵的相关结果.
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| 关 键 词： | 广义二次矩阵 广义多项式 不变性 换位子 |
| 收稿时间： | 2021-05-18 |
Rank Invariance of Generalized Polynomial of Generalized Quadratic Matrices |
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| CHEN Meixiang,YANG Zhongpeng,LIN Zhixing,FENG Xiaoxia.Rank Invariance of Generalized Polynomial of Generalized Quadratic Matrices[J].Journal of Jilin University: Sci Ed,2022,60(2):253-260. |
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| Authors: | CHEN Meixiang YANG Zhongpeng LIN Zhixing FENG Xiaoxia |
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| Institution: | 1. Key Laboratory of Applied Mathematics of Fujian Province University, Putian University, Putian 351100, Fujian Province, China；
2. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, Fujian Province, China |
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| Abstract: | Firstly, by using the relationship between generalized quadratic matrix A and idempotent matrix P, which was expressed as (A-dP)(A-eP)=0, we discussed the basic properties of generalized polynomial fP(A) of A, and proved the rank invariance of operations for generalized polynomials. The results show that ranks of generalized polynomials are not only independent of the choice of combination coefficients, but also independent of the choice of polynomials in most cases. Secondly, as applications, the relevant results of idempotent matrix, involutory matrix, quadratic matrix and generalized quadratic matrix were summarized and generalized. |
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| Keywords: | generalized quadratic matrix generalized polynomial invariance commutator |
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