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| 关于子块为矩阵多项式的矩阵的秩 | |
| 引用本文： | 刘英,王路群,刘冬丽. 关于子块为矩阵多项式的矩阵的秩[J]. 高师理科学刊, 2010, 30(5): 16-18. DOI: 10.3969/j.issn.1007-9831.2010.05.006 |
| 作者姓名： | 刘英 王路群 刘冬丽 |
| 作者单位： | 哈尔滨师范大学恒星学院信息科学系,黑龙江哈尔滨150025 |
| 基金项目： | 黑龙江省高教学会"十一五"规划项目 |
| 摘 要： | 为了进一步整合线性代数的内容,利用分块矩阵与λ-多项式理论对子块为矩阵多项式的矩阵的秩进行系统的论述.得到的主要结论:设B(λ)∈F[λ]s×t,A∈F n×n,则rank(B(A))=rank(h1(A))++rank(hr(A)),其中:r=rank(B(λ));h1(λ),,hr(λ)∈F[λ]为任意非零多项式,且h1(λ),,hr(λ)的标准分解式中不可约因子的方幂构成B(λ)的全部初等因子. |
| 关 键 词： | 矩阵的秩 λ-矩阵 不变因子 初等因子 |
The rank of partitioned matrices with polynomial matrix sub-blocks |
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| LIU Ying,WANG Lu-qun,LIU Dong-li. The rank of partitioned matrices with polynomial matrix sub-blocks[J]. Journal of Science of Teachers＇College and University, 2010, 30(5): 16-18. DOI: 10.3969/j.issn.1007-9831.2010.05.006 | |
| Authors: | LIU Ying WANG Lu-qun LIU Dong-li |
| Affiliation: | (Department of Information Science,Star College,Harbin Normal University,Harbin 150025,China) |
| Abstract: | In order to further integrate the content of linear algebra,revealed the rank of partitioned λ-matrices,in which every sub-block is matrix polynomial,using partitioned matrix andλ-polynomial theory.The main conclusion is that ifB（λ）∈F [λ ] s ×t,A∈F n ×n,thenrank（B（A））=rank（h1（A））＋ ＋rank（hr（A））.Where r=rank（B（λ））,h1（λ）,,hr（λ） ∈ F [λ] are any non-zero polynomial with the following properties that the powers of irreducible divisors in the canonical form of h1（λ）,,hr（λ） constitute all the elementary divisors ofB（λ）. |
| Keywords: | rank of matrix λ-matrix invariant factor elementary divisor |
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